VLBI with SKA: Possible Arrays and Astrometric Science

CH₃OH masers at 6.7 GHz were discovered by Menten (1991), which, along with 22.2 GHz H₂O masers, are the brightest and most widespread maser species in HMSFRs (e.g., Caswell et al. 1995; Minier et al. 2003; Xu et al. 2008). Sugiyama et al. (2013) conducted 6.7 GHz CH₃OH maser monitoring studies, using the EAVN to observe 36 target sources in HMSFRs. Fujisawa et al. (2014) subsequently released the results of this program, who found that maser emission was detected for 35 sources. This group published a detailed case study on the source G006.79-00.25, and found that 6.7 GHz CH₃OH maser spots were both rotating and expanding(Sugiyama et al. 2016). It is worth noting that Moscadelli et al. (2013) pointed out that 6.7 GHz maser emission traces gas entrained in the jet. In addition, Bartkiewicz et al. (2020) used the EVN to observe 6.7 GHz CH₃OH masers near the massive YSO G23.657-00.127, and concluded that it was hard to determine whether the methanol masers were tracing a spherical outflow arising from an (almost) edge on disk or a wide-angle wind at the base of a protostellar jet. SKA-VLBI will have high sensitivity and will be able to detect faint sources. Assuming an observation at 6.7 GHz with an integration time of 1 hr and a bandwidth of 2 kHz, a 1$\sigma$ would be possible at $\sim$1.4 mJy beam^-1 (see Table 3), which is better than the $\sim$4 mJy beam^-1 in Bartkiewicz et al. (2020). The estimated beam size ($\sim$0.9 mas) is also smaller than that in Bartkiewicz et al. (2020) by a factor of $\sim$7. These facts indicate it is very likely that more maser spots will be detected by SKA-VLBI, and the resulting astrometric precision will also improve. Combining search of radio continuum emission with high spatial resolution (such as by SKA-Mid) to pinpoint the driving source of the masers, SKA-VLBI is expected to discern between the scenarios of sphere-like outflow and wide angle wind at the base of the protostellar jet (see above, Bartkiewicz et al. 2020), i.e., distinguish which part of the outflowing gas near YSO is traced by 6.7 GHz CH₃OH masers. It is also possible to conduct further similar studies on more HMSFRs to determine statistically whether the scenario is unified for different sources.

Gould’s Belt Distances Survey (GOBELINS) was dedicated to measuring the distances and proper motions of young stars in neighboring molecular clouds using VLBA at 5 and/or 8 GHz with a bandwidth of 256 MHz. Results from this survey have been reported for the Ophiuchus complex (Ortiz-León et al. 2017a), Orion molecular clouds (Kounkel et al. 2017), the Serpens/Aquila molecular complex (Ortiz-León et al. 2017b), and the Perseus molecular cloud (Ortiz-León et al. 2018). The dynamics of star formation and young stars within molecular clouds were studied. For binary or multiple young stars, physical properties such as mass and orbital parameters have also been published (for more details see Section 3.2.1). The typical sensitivity of GOBELINS was $\sim$25 $\mu$Jy, which is far worse than the $\Delta S$ in Table 3, indicating that SKA-VLBI will be able to detect more younger stars (see also Ortiz-León et al. 2017a). Therefore, SKA-VLBI is expected to help compose a more complete picture of the synergistic study of young stars and molecular cloud dynamics.

In addition, it is interesting to note that in a case study of the Protostellar Outflows at the EarliesT Stages (POETS) survey presented by Sanna et al. (2019), linearly polarized emission from 6.7 GHz CH₃OH masers was detected for G035.02+0.35, and the direction of the magnetic field was found to be consistent with the axis of the jet. It is expected that studies of the Zeeman effect in 6.7 GHz CH₃OH masers (e.g., Vlemmings 2008; Crutcher & Kemball 2019; Dall’Olio et al. 2020) of early stage protostars will further complement studies of their magnetic fields by using SKA-VLBI with its higher sensitivity. This will enable the study of the magnetodynamics of SFRs and the influence of magnetic fields on star formation.

The significance of multiband and multiepoch observations of post-main-sequence stars (e.g., AGB stars, red supergiant stars, etc.) is that they can be used to (i) investigate the astrochemistry of material that has been ejected into the interstellar medium (ISM); (ii) distinguish different maser pumping mechanisms (e.g., radiative or collisional pumping), and derive these physical properties accordingly in circumstellar envelopes (CSEs); (iii) study the late stages of stellar evolution; and (iv) probe the spatial structure and dynamics related to mass-loss processes and measure asymmetry in CSEs (see Reid & Honma 2014; Rioja & Dodson 2020). The study of these post-main-sequence stars mainly relies on observations of 22.2 GHz H₂O, 43.1, 42.8, 86.2, and 129.3 GHz SiO, 1612, 1665, and 1667 MHz OH masers, and others, which are used to measure parallaxes, proper motions, radial velocities, and light variations, and to determine the overlaps or offsets between masers of different transitions, ring sizes, and their temporal evolution (Reid & Honma 2014). The VLBI parallaxes of AGB stars can also be compared with Gaia parallaxes, which are used to constrain the Gaia parallax zero-point offset (see Section 3.1.4 for details), and to calibrate the period-luminosity relation derived from oxygen-rich Mira variables in the MW (Andriantsaralaza et al. 2022).

Below 15 GHz, OH masers are good tracers of AGB stars. OH masers at 1612 MHz are a great tracer of low- and intermediate-mass oxygen-rich evolved stars with long-period stellar pulsations (i.e., types of AGB stars). This type of evolutionary stage with strong 1612 MHz OH maser emission is exclusively attributed to OH/IR stars. Observations of OH/IR stars can be traced back to the 1980s (e.g., Herman & Habing 1985; van Langevelde et al. 1990; Lindqvist et al. 1992), and include large-scale sky surveys performed with the Australian Telescope Compact Array (ATCA) and VLA (Sevenster et al. 1997a, b, 2001). Engels & Bunzel (2015) provided a database for querying known 1612, 1665, and 1667 MHz OH masers.⁸⁸8See https://hsweb.hs.uni-hamburg.de/projects/maserdb/. Combining the HI/OH/Recombination line (THOR) survey of the MW (Beuther et al. 2016, 2019) and follow-up observations by the Southern Parkes Large-Area Survey by the ATCA in Hydroxyl (SPLASH; Qiao et al. 2016, 2018, 2020), Uno et al. (2021) conducted 3D simulations of OH/IR stars traced by 1612 MHz OH masers in the Galactic disk. From these simulation data, and within a scale height of 150 pc, they found there may $\sim$3990 OH/IR stars in the Galactic plane, for which they determined their kinematics distances.

Ionospheric residuals can be largely reduced using MultiView, and a spatial resolution can reach $\sim$3.9 mas, thus rendering SKA-VLBI a suitable instrument for studying OH/IR stars traced by 1612 MHz OH masers. The specific technical methodology and scientific strategy for future SKA astrometric missions to study 1612 MHz OH masers is currently under development (see Orosz et al. 2014). Figure 7 shows the distribution of $\sim$3990 OH/IR stars, and the results show that SKA-VLBI can measure, in theory, the parallaxes of about 34% (1345/3990) of OH/IR stars with a distance precision of better than 10% (here, we have assumed a bandwidth of 500 Hz, an integration time of 1 hr, the astrometric properties in Table 3, the highest parallax precision of 6 $\mu$as, and the maximum Decl. (J2000) of 35^∘). OH masers at 1665 and/or 1667 MHz can also be used to study AGB stars (e.g., van Langevelde et al. 2000; Vlemmings et al. 2003; Vlemmings & van Langevelde 2007).

Deller et al. (2011) proposed a major project to measure the parallaxes of pulsars (some of which are in binary systems), PSR$\pi$, using VLBA. This project planed to observe 60 and 200 pulsars in its Phases I and II, respectively. Phase I was implemented over a 9 yr period (Deller et al. 2016, 2019) and the parallaxes and proper motions of 60 pulsars were obtained. The typical parallax precision was $\sim$45 $\mu$as, with the best being close to 10 $\mu$as, which was much more precise and accurate than pulsar’s dispersion measure (DM)-based distance (Chatterjee et al. 2009; Deller et al. 2009, 2019). It is worth noting that the astrometric errors of these near but weak sources are mainly determined by thermal noise (Rioja & Dodson 2020). Vigeland et al. (2018) reported a similar project, i.e., observing millisecond pulsars (which are highly probably in binary systems), MSPSR$\pi$, using VLBA, and the parallaxes and proper motions of 18 millisecond pulsars were reported (Ding et al. 2020, 2023).⁹⁹9For the specific status of PSR$\pi$ and MSPSR$\pi$, see https://safe.nrao.edu/vlba/psrpi/status.html.

In addition to measuring the parallaxes and proper motions of (millisecond) pulsars, a series of studies, including those related to projects PSR$\pi$ and MSPSR$\pi$, have made great progress obtaining the physical properties of pulsars, e.g., their effective temperature, radius, mass, and average electron density, and provided constrains on the time derivative of Newton’s gravitational constant (e.g., Vigeland et al. 2018; Deller et al. 2019; Ding et al. 2020, 2021, 2023). This series of work has also contributed to the development of pulsar timing array (PTA) studies, such as providing accurate distances to pulsars, improving PTA sensitivity, and verifying and improving pulsar timing solutions (Madison et al. 2013; Vigeland et al. 2018; Deller et al. 2019). At the same time, pulsars provide a rich laboratory for studying nuclear and particle physics (Lorimer 2008; Deller et al. 2011); they can be used to test GR (Deller et al. 2009, 2018; Ding et al. 2021; Guo et al. 2021); they were used to find the first observational evidence for the existence of gravitational waves (Taylor & Weisberg 1989); and their study led to the discovery of the first exoplanet (Wolszczan & Frail 1992).

The advantages of MultiView are required for determining precise astrometry for pulsars, so that stronger sources with larger angular distances can be used as reference calibrators, providing commensurate precision (see Rioja & Dodson 2020). The high sensitivity of SKA-VLBI not only will enable the detection of weak sources while maintaining high DR, but its will also enable in-beam MultiView observations. In this section, the pulsar astrometry presented here is based on the assumption that astrometric precision is dominated by thermal noise. Herein, we consider the pulsar catalog provided by Manchester et al. (2005) and recently updated by (Manchester et al. 2016), which contains 2536 pulsars. (Note that the pulsars published in PSR$\pi$ and MSPSR$\pi$ are also in this catalog.) Figure 8 shows the distribution of these pulsars, of which 12% (295/2536) have parallax and/or proper motion measurements. In the catalog of Manchester et al. (2016), 1671 pulsars have flux density measurements (at 1.4 GHz), and 2481 have distance measurements (default as DM-based distance), but 864 have no flux density or distance measurements. According to the information in the catalog and the astrometric properties listed in Table 3, in theory, a total of $\sim$48% (1225/2536) of these pulsars, located with a Decl. (J2000) below 35^∘, can be measured with a distance precision better than 10%, and sources can be detected by to 12.5 kpc (where the highest parallax precision is assumed to be $\sim$8 $\mu$as, i.e., 1/3 of the astrometry precision listed in Table 3). There are 1014 pulsars whose flux densities are higher than the sensitivity matched to the DRs of 100:1 (i.e., $\geq$ 0.27 mJy beam^-1) and Decl. below 35^∘. Note that the observed DRs of pulsars can be enhanced when pulsar gating is applied in the correlator (see Deller et al. 2019; Ding et al. 2023). It is undeniable that SKA-VLBI astrometry with high sensitivity and high precision will be able to observe a large number of pulsars, making contributions to the measurement of electron density in the MW and the other science goals mentioned above.

Radio stars are stars visible in the radio band, which contain young stars, AGB stars, pulsars, etc. Multiepoch SKA-VLBI observations of radio stars can yield both high-precision astrometric parameters (such as parallax and proper motion) and high spatial resolution images. Combined with optical Gaia astrometry, the high-precision astrometry of SKA-VLBI can be used to link the optical and radio celestial reference frames, and verify the astrometric performance of VLBI and Gaia.

Long-term astronomical observations in both radio and optical bands can produce celestial reference frames, such as ICRF3 (Charlot et al. 2020) for radio and the Gaia DR2/(E)DR3 celestial reference frame (Gaia-CRF2/CRF3, which is collectively called GCRF; Gaia Collaboration et al. 2018, 2022) in the optical. As mentioned previously, there are also notable VLBI-Gaia positional offsets (Mignard et al. 2016; Kovalev et al. 2017; Petrov & Kovalev 2017; Charlot et al. 2020; Liu et al. 2021; Secrest 2022). Observations of radio stars are helpful to tie together these international celestial reference frames (Reid & Honma 2014; Malkin 2016, for more details see Section 3.3.3).

It is interesting to compare the parallaxes of evolved AGB stars and other radio stars obtained with VLBI (i.e., such as the more than 130 stars used in the study by Xu et al. 2019) with those obtained with Gaia, as such as comparison offers a unique opportunity to test the Gaia parallax zero-point offset. Xu et al. (2019) compared the VLBI and Gaia parallaxes of their sample of stars and found that except for AGB stars, there was a systematic offset between the parallaxes, $-75\pm 29$ $\mu$as, which was consistent with the zero-point offset of Gaia. For AGB stars, there was a significant difference between the Gaia and VLBI parallaxes. Andriantsaralaza et al. (2022) compared the VLBI parallaxes of AGB stars with those of Gaia DR3 and obtained a zero-point offset of DR3 parallax of $\sim-77$ $\mu$as for bright AGB stars. They also found that the zero-point offset become more negative for fainter AGB stars, and Gaia DR3 underestimated the uncertainties of the parallaxes of these AGB stars.

Radio star astrometric observation plans were made in the latter part of the last century (e.g., Lestrade et al. 1986; Garrington et al. 1995; Lestrade et al. 1999; Johnston et al. 2003) and the early part of this century (e.g., Boboltz et al. 2003, 2007). Wendker (1995) provided a catalog of radio stars (note that the sample of radio stars discussed in this passage specifically refers to stars from this catalog) which is continually updated and currently contains 3699 stars (their distribution is displayed in Figure 9). These radio stars were observed at frequencies ranging from 20 MHz to 420 GHz, with 3230 stars falling within the SKA-VLBI frequency range (which will be given priority in this work). SKA-VLBI will be able to measure precisely the parallaxes of radio stars whose flux densities (or upper limits) are higher than the sensitivity matched to the DR of 100:1 (only considering sources observed at 1.0–2.0 and 4.6–15.3 GHz and had a Decl.(J2000) $<$ 35^∘). The number of such radio stars is $\sim$2033, which is about one order of magnitude higher than the number of radio stars that have reported VLBI parallaxes.

Matthews et al. (2023) reported the detection of 15 GHz continuous emission for the first time using VLA observations of a classic Cepheid, $\delta$ Cephei ($\delta$ Cep; distance of $\sim$0.3 kpc), with a flux density $\sim$15 $\mu$Jy. For an integration time of 4 hr and a bandwidth of 1 GHz, the corresponding sensitivity of SKA-VLBI is $\sim$1.8 $\mu$Jy beam^-1. SKA-VLBI has the potential for direct triangular parallax measurements of classical Cepheids to be conducted, allowing astronomers to calibrate the period-luminosity relation more precisely. The theoretical distance precision is better than 10% for a classic Cepheid within 1.0 kpc assuming that its flux density at 0.3 kpc is $\sim$15 $\mu$Jy at 15 GHz (as in the case of $\delta$ Cep), where the assumed flux density is inversely proportional to the square of distance. Pietrukowicz et al. (2021) cataloged classical Cepheids in the MW, updated as of 2023,¹⁰¹⁰10See https://www.astrouw.edu.pl/ogle/ogle4/OCVS/allGalCep.listID. and these classical Cepheids have been cross-matched with Gaia DR3 (Gaia Collaboration et al. 2023). There are over 3000 classical Cepheids in this catalog, with $\sim$80 within $\sim$1.0 kpc. Ripepi et al. (2023) provided a Gaia DR3 catalog of Cepheids containing some in neighboring galaxies, totaling over 15,000 Cepheids.

Combining high-precision VLBI astrometry and lunar ranging data, the mass of the Earth-Moon system and the Moon’s momentum of inertia ratio can be measured (see Lanyi et al. 2007; Reid & Honma 2014; Reid et al. 2019b). Combining VLBI data with optical observations, Doppler ranging data, and data from the Cassini satellite, the mass and potential of the Saturnian system were constrained (e.g., Jacobson et al. 2006). In addition, high-precision VLBI astrometry can also be used to monitor the trajectories of artificial radio sources (such as man-made spacecraft), contribute to geophysical research, (such as measuring high-altitude winds, see Counselman et al. 1979; Sagdeyev et al. 1992; Reid & Honma 2014), and improve future space missions (see Duev et al. 2012). More research on satellite tracking can be found in the review by Reid & Honma (2014).

Binary or multiple stellar systems include young binary or multiple stars, X-ray binaries (where one of the stars is a stellar black hole, a neutron star, or a white dwarf), Hulse-Taylor binary pulsar systems (HTBPSRs, which are composed of a neutron star and a pulsar, see Hulse & Taylor 1975), and other binary or multiple stellar systems containing neutron stars or black holes, cataclysmic variables (CVs, where the primary star is a white dwarf); see more details below. It is possible to monitor the orbital motions and measure the parallaxes, masses, orbital periods, and other physical quantities of these systems. The significance of studying binary or multiple stars includes testing theories of stellar structure and evolution; exploring star formation mechanisms and environmental effects in the galactic gravitational potential and in clusters; understanding important types of astrophysical phenomena such as Type Ia supernovae, CVs, and stellar X-ray sources (Horch 2013); and testing fundamental physical theories such as the relativistic effects of binary/multiple stellar systems containing neutron stars and/or black holes (e.g., Deller et al. 2018; Ding et al. 2021; Guo et al. 2021); etc.

Younger binary or multiple stars. Orbital monitoring of young binary stars using VLA/VLBA has been carried out (e.g., Curiel et al. 2002; Loinard 2002; Rodríguez et al. 2003; Maureira et al. 2020, and GOBELINS, see Section 3.1.1). These studies provided orbital and other physical parameters of the binary/multiple-star systems. For example, Rodríguez et al. (2003) obtained the total mass and orbital period of the binary L1551 IRS 5 using the VLA, finding $\sim$1.2 $M_{\odot}$ and $\sim$260 yr, while Maureira et al. (2020) combined VLA observations from Loinard (2002) and ALMA data to derive the mass range of the binary IRAS 16293-2422 A, finding $0.5\lesssim M_{1}\lesssim M_{2}\lesssim 2\;M_{\odot}$. Using VLBA with its higher astrometric precision, GOBELINS also monitored the orbits of 17 young binary and nine young multiple stars, and obtained the orbital parameters for a total of 12 binary stars, including the mass of each single star in the binary system for nine systems. The maximum precision reached $\sim$0.02 $M_{\odot}$ (corresponding to a relative precision up to better than 10%; see Kounkel et al. 2017; Ortiz-León et al. 2017a, b, 2018). Among them, Ortiz-León et al. (2017a) even determined the mass of two stars in a triple-star system, YLW 12B. The value of $\theta_{\mathrm{beam}}$ at 5.0 GHz is $\sim$1.2 mas (see Table 3), and the astrometric precision is better than 11 $\mu$as (assuming the flux density at 5 GHz is better than the minimum continuous emission of $\sim$0.2 mJy in GOBELINS; Ortiz-León et al. 2017a, i.e., the matched DR is $\sim$53:1, as shown in Table 3). Both of these are less than the semimajor axes of binary younger stars reported in GOBELINS (i.e., over $\sim$10 mas). Therefore, the astrometric precision that may be obtained using SKA-VLBI may be sufficient to conduct orbital monitoring of such binary and multiple star systems and more (see Section 3.1.1).

X-ray binaries. Some X-ray binaries have properties similar to quasars or other AGNs, but as they evolve much faster, they are often called micro-quasars. X-ray binaries can be used to study the interaction of stars with the accretion disks of the black holes or neutron stars they orbit, and to study the stability and lifetime of the corresponding jets; such approaches have been useful for studying the relativistic jets of distant quasars (Mirabel & Rodríguez 1998; Reynolds 2021). An important example of VLBI observation of X-ray binary is Cygnus X-1 (Reid et al. 2011; Orosz et al. 2011; Miller-Jones et al. 2021); after obtaining the distance to this X-ray binary and monitoring its orbit, the mass-distance degeneracy in the simulation of optical/infrared data was broken, and the mass of the black hole and its companion star was therefore obtained. In addition, the parallaxes of X-ray binaries obtained with VLBI can be compared with the distances obtained from optical observations, which can be used to constrain theories such as accretion disk theory. For example, the distance to SS Cyg measured by EVN and VLBA (Miller-Jones et al. 2013) was smaller than that determined using the Hubble Space Telescope observations (Harrison et al. 1999), making the distance more consistent with accuracy disk theory (for a review see Reid & Honma 2014). Several examples of X-ray binaries with measured parallaxes include Sco X-1, V404 Cyg, Cygnus X-1, and SS Cyg (see Fomalont et al. 2001; Miller-Jones et al. 2009; Reid et al. 2011; Miller-Jones et al. 2013), which were observed at frequencies of 1.7–22.2 GHz and their measured flux densities were $\gtrsim$ 5 mJy beam^-1. It is expected that SKA-VLBI will be able to measure the parallaxes and semimajor axes of primary stars with a theoretical parallax precision of $\sim$1 $\mu$as. It is also expected that SKA-VLBI will enable the exploration of fainter X-ray binaries and allow astronomers to monitor their light curves (e.g., Ng et al. 2023) with high sensitivity. Samples of X-ray binaries can be obtained from the Galactic high mass X-ray binary catalogue (Neumann et al. 2023) and the Galactic low mass X-ray binary catalogue (Avakyan et al. 2023).

HTBPSRs. An HTBPSR is a specific type of binary containing a pulsar (Hulse & Taylor 1975). Such binaries led to the first indirect detection of gravitational waves (see Taylor & Weisberg 1982, 1989) and can also be used to test GR (see Section 3.4.2 for details). An important representative is pulsar B1913+16, which has been studied extensively for many years. Weisberg & Taylor (2005) made an important review of studies of B1963+16, and since then there have been observations conducted by large radio telescopes and arrays (e.g., Arecibo and VLBA; Nice et al. 2005; Deller et al. 2018) and analyses of long-term observational data (e.g., Weisberg & Huang 2016). Weisberg & Huang (2016) obtained the ratio of the observed to GR-predicted orbital period decay rate (ROPR hereafter) of B1913+16 due to gravitational wave damping, $0.9983\pm 0.0016$, which tested the GR with an precision of 0.2%. They also found that the result was also consistent with the GR-predicted value from the Shapiro gravitational propagation delay. Using eight epochs of observations made over 2 yr with VLBA, Deller et al. (2018) obtained the ROPR, finding a value of $1.005^{+0.001}_{-0.003}$, which was incongruous with that of Weisberg & Huang (2016) but with similar level of precision. Therefore, further radio VLBI observations with higher astrometric precision than achieved for pulsar B1913+16 and observations toward more HTBPSRs (which may be fainter) are imperative. These, in conjunction with pulsar observations (see Section 3.1.3), will constitute an important area of focus for SKA-VLBI.

Cataclysmic variables. CVs are interacting binaries in which a white dwarf accretes material from a low-mass companion. CVs are valuable to test theories of the evolution of binary stars; however, it is difficult to obtain precise values of the masses and radii of each star in a CV (Knigge et al. 2011; Littlefair et al. 2024). VLBI detections of CVs are scarce (e.g., nova RS Ophiuchi by VLBA; O’Brien et al. 2006, see nova below). Recently, Jiang et al. (2024) reported an EVN detection of an AE Aquarii (AE Aqr)-type CV,¹¹¹¹11It is a subtype of intermediate polars, which are a subclass of CVs with a moderate magnetic field strength of $10^{5}\lesssim B\lesssim 10^{7}$ G and a sufficiently fast spin rate. LAMOST J024048.51+195226.9, which possesses the fastest known rotating white dwarf with a spin period of $\sim$25 s. This paper reported only its equatorial coordinates and no other astrometric parameter. Its maximum flux density at 1.7 GHz was $\sim$1.0 mJy, indicating a potential target for SKA-VLBI to explore.

Novae. A nova is also a kind of CV. A classical nova is an eruption event on the surface of an accreting white dwarf binary. In such an event, the typical released energy is $\sim$10³⁸–10⁴³ erg, and heavy elements inside a white dwarf are ejected into the ISM (Gallagher & Starrfield 1978; Gehrz et al. 1998; José & Hernanz 2007; Romano et al. 2017), implying novae have a significant impact on the ISMs. Currently, theory and observations of novae are not overly consistent, especially in terms of the ejecta masses of different subtypes of novae. Radio observations are an ideal tracer of ejecta masses, and have already made significant contributions to the study of novae, such as constraining the expanding ejecta and their morphologies, distances, shell masses, and kinetic energies (see the short review by Gulati et al. 2023). Gulati et al. conducted a systematic search for novae with Decl. $\leq$ 41^∘ using ASKAP, discovering four nova radio counterparts, and conducted follow-up observations using ATCA. Their results highlight ASKAP’s ability to contribute to future radio research of novae. The maximum radio flux density of a nova is related to the ejected mass, $M_{\mathrm{ej}}$. At 1 GHz, assuming $M_{\mathrm{ej}}\sim 5\times 10^{-4}$ $M_{\odot}$, SKA-VLBI may, in theory, be able to locate sources within $\sim$10 kpc with an precision of better than 10% (based on the astrometry in Table 3) and the maximum flux density, which is associated with the $M_{\mathrm{ej}}$ taken from table 4 in Gulati et al. 2023).¹²¹²12For example, the maximum flux density at 1.0 GHz for a novae at 0.8 kpc is $\sim$ 37.71, $\sim$ 5.98 and $\sim$ 0.95 mJy for $M_{\mathrm{ej}}\sim 5\times 10^{-4}$, $\sim 5\times 10^{-5}$ and $\sim 5\times 10^{-6}$ $M_{\odot}$, respectively. Note that the assumed maximum flux density is inversely proportional to the square of the distance. If the values of $M_{\mathrm{ej}}$ are $5\times 10^{-5}$ and $5\times 10^{-6}$ $M_{\odot}$, the corresponding distance becomes $\sim$6 and $\sim$3 kpc, respectively. In these cases, the corresponding mass of the white dwarf is $\sim$0.4–1.25 $M_{\odot}$ (Yaron et al. 2005; Gulati et al. 2023).

Electromagnetic counterparts to gravitational wave sources. Such sources correspond to merge of two neutron stars (NSNS), or a neutron star and a black hole (NSBH). A correlation linking short gamma-ray bursts, kilonovae, and NSNS/NSBH merges can be found in figure 1 of Fernández & Metzger (2016). Alexander et al. (2017) detected the electromagnetic counterpart of the binary neutron star merger detected by LIGO/Virgo, GW170817 (Abbott et al. 2017; LIGO Scientific Collaboration & Virgo Collaboration 2017) at 6–15 GHz with a flux density of several 100 $\mu$Jy beam^-1. A superluminal motion of a relativistic jet with flux density of $\sim$50 $\mu$Jy beam^-1 at 4.5 GHz was subsequently reported by Mooley et al. (2018). The expected sensitivity of SKA-VLBI will reach several $\mu$Jy beam^-1 (see Table 3) for an integration time of 1 hr at those frequencies, where the physical scale corresponding to $\sim$0.8 mas at 100 Mpc is $\sim$1 pc. Therefore, SKA-VLBI will be able to make movies of expanding ejecta or jets of merging NSNSs or NSBHs. SKA-VLBI is also expected to witness the precession of supermassive black holes (SMBHs), and provide a unique way to obtain a resolved image of the electromagnetic counterpart of a gravitational wave event (Reid et al. 2019b).

Other binaries. In this section we consider a binary star 1.8 kpc away that is similar to IM Pegasi (IM Peg; HR 8703; see Ransom et al. 2012; Fomalont & Reid 2004). The semimajor axis of the primary star is $\sim$0.1 AU, the orbital period is $\sim$25 days, and the flux density at 8.4 GHz is typically $\sim$10 mJy at a distance of 96.4 pc. In such a case, the astrometric precision would reach about 57 $\mu$as (where the flux density is assumed to be 29 $\mu$Jy, i.e., inversely proportional to the square of the distance), corresponding to an angular radius of $\sim$0.1 AU, which is similar to the assumed semimajor axis of the primary star. Therefore, it is expected the SKA-VLBI will provide high-precision astrometry of binary star orbits, at least for binaries within a distance of $\sim$1.8 kpc. Table 5 in Schwarz et al. (2016) lists catalogs of binary stars, and Mason et al. (2001) cataloged over 100,000 binary stars. Radio stars or OH/IR stars can also be sources of binary stars.

The first exoplanets were discovered when precise timing measurements were conducted of the 6.2 ms pulsar PSR1257$+$12 using the Arecibo radio telescope (Wolszczan & Frail 1992). In that work, the pulsar was thought to be orbited by two or more planet-sized bodies with masses of at least 2.8 and 3.4 $M_{\bigoplus}$ (where $M_{\bigoplus}$ is the mass of Earth). Searching for exoplanets using VLBI dates back to the work of Lestrade et al. (1996), where observations over a 7.5 yr period excluded a Jupiter-mass planet orbiting at a radius of 4 AU from the central star $\sigma^{2}$ CrB, with postfit residuals of $<$0.3 mas (Reid & Honma 2014). Subsequently, Guirado et al. (1997) observed the active K0 star AB Dor and their result indicated a companion with a mass of $\sim$0.1 $M_{\odot}$. Two projects which focused planets and dwarfs, respectively, were subsequently conducted: the Radio Interferometric Planet (RIPL) Search (Bower et al. 2009) and the Radio Interferometric Survey of Active Red Dwarfs (RISARD, Gawronski et al. 2013). The RIPL collaboration reported on studies of GJ 896A, and found that the upper limit of the companion of GJ 896A was 0.15 $M_{J}$ (where $M_{J}$ is the mass of Jupiter) with a semimajor axis of 2 AU, where the precision of the parallax and proper motion were 0.2 mas and 0.01 mas yr^-1, respectively (Bower et al. 2011).

Recently, Curiel et al. (2022) found that the mass and the orbit period are 2.3 $M_{J}$ and 284.4 days, respectively, for the single planet of the main star (GJ 896A) of the binary, GJ 896AB, through orbit monitoring with VLBA, and optical and infrared data. However, it is still poorly understood for outer planetary systems (Reid et al. 2019b). Fortunately, Schwarz et al. (2016) cataloged binary or multiple star system containing exoplanets,¹³¹³13See https://adg.univie.ac.at/schwarz/multiple.html. with a total of 217 candidates, and the Extrasolar Planets catalog¹⁴¹⁴14See http://exoplanet.eu/catalog/. also lists 4000 planetary systems (Butler et al. 2006; Le Sidaner et al. 2007; Schneider et al. 2011). These objects can serve as pilot target sources for investigating exoplanets and potential dwarfs by using SKA-VLBI, which has a higher sensitivity than VLBA.

The nearest SMBH is Sgr A^∗ at the center of the MW. Monitoring pulsar orbits around Sgr A^∗ with precision astrometry has allowed astronomers to measure and test GR effects (although the observational frequency needs to be greater than about 10 GHz due to interstellar scattering; Reid & Honma 2014). Reports of monitoring observations of pulsars near the GC include only PSR J1745-2900 (see Mori et al. 2013; Bower et al. 2015). It is an opportunity for SKA-VLBI to conduct orbital monitoring of more pulsars to test GR with high precision. In addition, measuring the proper motion of Sgr A^∗ will allow astronomers to constrain the mass of Sgr A^∗, evaluate the Sun’s orbital motion around the GC, such as its circular rotation speed, $\Theta_{0}$, and constrain the mass density of SMBH candidates (see Reid et al. 1999; Reid & Honma 2014; Reid & Brunthaler 2020).

The distances to star clusters can also be determined by using VLBI to measure parallaxes. Although the emission from individual stars in a cluster is very weak, the sensitivity of SKA may allow astronomer to observe them (Fomalont & Reid 2004). At present, the famous and the only example of using VLBI to measure the distance of a star cluster is the Pleiades cluster, where Melis et al. (2014) measured a distance of 136.2 $\pm$ 1.2 pc, which suggested that the Hipparcos-measured distance to the Pleiades cluster (i.e., 120.2 $\pm$ 1.5 pc) was in error. At the same time, Melis et al. (2014) proposed that Gaia should identify the unrecognized error in the Hipparcos measurement and correct it. Indeed, Gaia DR2 measured a distance of 135.15 $\pm$ 0.43 pc, which ended the “Pleiades distance controversy” (Lodieu et al. 2019). The flux of member stars in the Pleiades cluster is around several 100 $\mu$Jy at $\sim$8 GHz (see Melis et al. 2014). Assuming a flux density of 100 $\mu$Jy, SKA-VLBI can also measure the parallax of this cluster with a theoretical precision of $\sim$7$\times 10^{-4}$, i.e., a distance error of $\sim$0.1 pc, where the adopted integration time and bandwidth are shown in Table 3. The high-sensitivity SKA-VLBI has the potential to measure precisely the parallaxes to many more stellar clusters hosting radio stars (see Section 3.1.4), and compare them with those determined by Gaia.

From a few detections of 6.7 GHz CH₃OH masers in external galaxies (only detected in the Large Magellanic Cloud and M31, and tentatively detected in Maffei 2), it has been found that they are likely located in star-forming regions (clusters) and have peak luminosities in the range $\sim$500–$2.3\times 10^{4}$ Jy kpc² (Chen et al. 2022). Assuming a peak luminosity of $10^{4}$ Jy kpc², the peak flux density of a 6.7 GHz CH₃OH maser 100 kpc away can reach $\sim$1 Jy. Therefore, it is expected that SKA-VLBI can be used to measure precisely the parallaxes and proper motions of external galaxies within 60 kpc using 6.7 GHz methanol masers as tracers (assuming a parallax precision of $\sim$1.7 $\mu$as, i.e., 1/3 of the astrometric precision shown in Table 3). If the parallax precision is $\sim$1/5 of astrometric precision (which is achievable), the distances of external galaxies whose parallaxes can be precisely measured can reach even $\sim$100 kpc.

For a more distant galaxy, its distance can be derived by measuring the proper motion of two or more target sources within it, and comparing them with the galaxy’s rotation model after obtaining its inclination angle, which is known as the van Maanen experiment (van Maanen 1923; Fomalont & Reid 2004; Reid & Honma 2014). The distance to M33 (which is about 730 kpc) was measured using this method (Brunthaler et al. 2002, 2005, 2008). Two distance galaxies, M31 and Maffei 2 (with distances of $\sim$0.8 and $\sim$5.7 Mpc, respectively), have (tentatively) detected 6.7 GHz CH₃OH masers (see Sjouwerman et al. 2010; Chen et al. 2022). Therefore, 6.7 GHz CH₃OH masers may be a potential tracer to measure distances via the van Maanen experiment. Such measurements of proper motions and geometric distances of galaxies within the Local Group are helpful to understand the Local Group’s history, present state and future (Brunthaler et al. 2008). By applying the measured distances of external galaxies using the van Maanen experiment to distances of standard candles (e.g., Cepheids, Type Ia supernovae, etc.) inside them, the accuracy of the cosmic distance ladder may be improved to 1% or better (Fomalont & Reid 2004), and thus improve the accuracy of $H_{0}$. The proper motions of galaxies can also be used to study galaxy evolution; for example, it may be possible to predict whether will M 31 and its two satellite galaxies M 33 and IC 10 collide with the MW in the future (e.g., Reid et al. 2019b).

Determining the structure of the MW has long been a difficult task. In the past 20 years, advances in astrometry have led to a clearer understanding of the structure of the MW, especially within 5 kpc of the Sun. Such progress is based on high-precision radio astrometry applied to obtain the triangular parallaxes of masers in HMSFRs, and optical Gaia parallaxes of neighboring stars and open clusters (Xu et al. 2018a; Reid et al. 2019a; Hao et al. 2021; Hou 2021; Xu et al. 2021a). Nevertheless, to this day, controversy remains regarding such basic facts as the existence of some spiral arms, the number of arms, and the size of the MW. The mainstream view of the structure of the MW is that it is a four-arm spiral galaxy extending from the GC to the outer regions, with some extra arm segments and spurs (see Georgelin & Georgelin 1976; Reid et al. 2019a). Xu et al. (2023) proposed a new paradigm where the MW has a multiple-arm morphology with two symmetrical arms in the inner parts and is split into four spiral arms through two nearly symmetry bifurcations (see Figure 10). This new model suggests the MW may be an ordinary galaxy, as only 2% of known multiple-arm galaxies have four inner arms, while 83% of have only two inner arms (Xu et al. 2023).

The most effective solution to addressing the controversy regarding the spiral structure of the MW is to obtain more parallax measurements of masers in HMSFRs widely distributed in the MW, including in the southern sky and to regions beyond the GC, as well as improving the parallax precision of objects. Compared with Gaia, maser emission is not absorbed significantly by interstellar dust, and is an excellent tracer for the entire MW. SKA-Mid will be located in the Southern Hemisphere. SKA-VLBI can cover regions of the sky with Decl. (J2000) $<$ 35^∘ with a theoretical parallax precision up to $\sim$1.7 $\mu$as, which can compensate for the current scarcity of triangular parallax measurements of masers in the southern sky.

Figure 10 shows the distribution of 6.7 GHz CH₃OH masers¹⁵¹⁵15The tracer of the spiral arm structure shown in Figure 10 is the 6.7 CH₃OH maser, because surveys of 12.2 GHz CH₃OH masers are mostly conducted toward regions where 6.7 GHz CH₃OH masers were detected, while 22 GHz H₂O masers are not considered in this work. superposed on the newly determined multiple-arm structure of the WM. The black points indicate 91 masers that currently have a parallax measurement made by the BeSSeL survey (Reid et al. 2019a; Xu et al. 2021b; Bian et al. 2022). The red points represent masers that can be observed by SKA-VLBI (i.e., Decl. (J2000) $\leq$ 35^∘), which include the catalog of Xu et al. (2009) and several other catalogs (with a large number of newly discovered masers) from 2008 onward, including the southern methanol maser sources observed by ATCA (Caswell 2009), the subsequent Methanol MultiBeam (MMB) survey (Caswell et al. 2010; Green et al. 2010; Caswell et al. 2011; Green et al. 2012; Breen et al. 2015), the Torun catalogue (Szymczak et al. 2012), and the survey conducted with the Tianma telescope (Yang et al. 2019). There are many masers located near the two bifurcations in the model of Xu et al. (2023), which would be a key sample to differentiate between this model and that of Reid et al. (2019a). To summarize, from this figure, SKA-VLBI is an ideal instrument to solve the controversy over the spiral structure of the MW (see above), and to map the relatively complete structure of the MW.

By comparing the spiral arm structures traced by masers with those traced by OB stars and open clusters obtained from Gaia astrometry, it is possible to study the consistency of the spiral arm structures traced by different young objects and the homogeneity of the distribution of star formation in the spiral arms (e.g., Xu et al. 2018b, 2021a). Naturally, we also expect SKA-VLBI to be able to observe the spiral arm structure depicted by other tracers, such as AGB stars (see Section 3.1.2), which can be compared with the results from Gaia astrometry to study the spiral arm structure traced by old objects (e.g., Poggio et al. 2021; Lin et al. 2022). It is also interesting to compare the spiral structure traced by old and young objects (e.g., Lin et al. 2022).

The parallaxes and proper motions of masers can also be used to study the structure, motion, and dynamics of spiral arms and the bar (e.g., Immer et al. 2019; Reid et al. 2019a; Li et al. 2022a); the dynamics in the central molecular zone (i.e., the central few 100 pc from the GC; e.g., Chambers et al. 2014; Kruijssen et al. 2015; Rickert 2017) and in the circumnuclear disk (e.g., Karlsson et al. 2003; Sjouwerman & Pihlström 2008) of the MW; and to measure the fundamental parameters of the MW. These parameters include $R_{0}$, $\Theta_{0}$, the height of the Sun from the Galactic plane, $Z_{\odot}$, and the peculiar motion of the Sun along the direction of its rotation, toward the GC, and the north pole, ($U_{\odot}$, $V_{\odot}$, $W_{\odot}$), (e.g., Reid et al. 2014; Xu et al. 2018a; Reid et al. 2019a). Note that $R_{0}$ is a key parameter that directly affects measurements of the mass and luminosity of almost all objects in the MW (Bland-Hawthorn & Gerhard 2016). In addition, these parallaxes and proper motions can be used to derive the rotation curve of the MW, thus giving kinematic distance models (Reid et al. 2014, 2019a; Reid 2022). 3D motion in the vicinity of the solar system can be studied by combining the parallaxes and proper motions of HMSFR masers measured by VLBI and OB stars and young open clusters by Gaia, which produce local velocity field and improve kinematics distance models (Liu et al. 2023).

The expansion or contraction of the universe directly manifests itself as a change in the relative position between pairs of galaxies or galaxy clusters. Therefore, measuring the proper motions of galaxies or galaxy clusters can directly determine whether the universe is expanding or even accelerating, and subsequently the Hubble constant and the distance of galaxies (Darling 2013). For distant galaxies, the ideal target sources are AGNs. By measuring the proper motions of bright radio source pairs, Darling (2013) obtained inclusive $8.3\pm 14.9$ $\mu$as yr^-1 for small separation pairs ($<200$ Mpc, $z<$ 0.1, where $z$ is the redshift), which was not constraining enough to determine whether the universe was Hubble expanding or static. However, for large separation pairs (200–1500 Mpc), they obtained no net convergence or divergence for objects with $z<1$, finding $-2.7\pm 3.7$ $\mu$as yr^-1, which is consistent with pure Hubble expansion and significantly inconsistent with a static universe. In order to determine further whether the relative proper motions of source pairs at distances $<200$ Mpc and $z<0.1$ support Hubble expansion, more source pairs observed with higher astrometric precision and multiple observations of target sources are required. SKA-VLBI astrometry would be helpful to solve this issue.

Other applications of proper motion measurements of galaxies include measuring the secular aberration drift caused by the finite speed of light and the motion of the observer with respect to a light source; secular parallax, which is the motion of the solar system barycenter with respect to the cosmic microwave background (CMB) rest frame; the rotation of the reference frame; the anisotropic expansion of the universe; gravitational waves; large-scale structure; and baryon acoustic oscillation evolution; for more details see Darling et al. (2018). Secular aberration drift, aberration proper motion fields, and rotation are tightly tied to the ICRF, and affect the astrometry to all objects (Titov et al. 2011; Truebenbach & Darling 2017; MacMillan et al. 2019; Charlot et al. 2020).

Moreover, the peculiar velocity field on large scales at low redshift is a unique tracer to probe the dark matter density field, where peculiar velocities can be separated from recessional velocities due to the Hubble flow by using directly measured distances (Boruah et al. 2020). It is feasible to measure the distances to nearby galaxies with SKA-VLBI (see Section 3.2.4), and such measured distance can be used to determine the distances to Cepheids and Type Ia supernova inside those galaxies, and thus improve the precision of the cosmic distance ladder. It is also possible to obtain geometrical distances of extragalactic objects by using a large sample of extragalactic proper motions (Kardashev 1986; Ding & Croft 2009). In this method, it was suggested to use Earth’s motion with respect to the CMB, which is $\sim$800 AU over a 10 yr period, as a baseline to measure the proper motions of extragalactic objects. Assuming a positional precision of 2 $\mu$as, 10 yr of monitoring would achieve a theoretical precision of $\sim$0.2 $\mu$as yr^-1, and a measurable distance with an uncertainty of 20% could be reached for an object $\sim$80 Mpc distant with a baseline of 800 AU using SKA-VLBI (see Figure 11). Chasing the 1 $\mu$as limit is exciting, as it can lead to measurable distances of galaxies as high as $\sim$160 Mpc.

Absolute astrometry allows one to determine the true position of a radio source defined in a well-defined inertial frame (Fomalont & Reid 2004), e.g., the ICRF. The ICRF is a key problem in fundamental astrometry.

Since 1995, the VLBI-based ICRF (i.e., ICRF1, Ma et al. 1998) has been adopted as the IAU standard ICRF, including its second and third versions, ICRF2 and ICRF3, respectively (Fey et al. 2015; Charlot et al. 2020). At the same time, the optical celestial reference frame has also been constantly updated, such as the Hipparcos celestial reference frame (HCRF), and its successor, the GCRF based on Gaia DR2/(E)DR3 (Gaia Collaboration et al. 2018, 2022). Many comparisons between the VLBI-based ICRF and optical celestial reference frame have been made, resulting in the detection of VLBI-Gaia positional offsets (see Section 3.1.4). Among them, the noise floor of ICRF3 is $\sim$30 $\mu$as for individual source coordinates (Charlot et al. 2020), and the typical positional precision of sources in GCRF is $\sim$1 mas (Gaia Collaboration et al. 2022).

On the one hand, it is possible to solve the VLBI-Gaia positional offsets by improving the astrometric precision and spatial resolution of radio and optical observations, as well as obtaining more multiband observations and long-term monitoring of ICRF-defining sources. On the other hand, Froeschle & Kovalevsky (1982) proposed using radio stars to link the VLBI-based ICRF with optical celestial reference frames (e.g., HCRF and GCRF; see Kovalevsky et al. 1997; Malkin 2016). This methodology is constantly improving and has achieved fruitful results (e.g., Malkin 2016; Lindegren 2020; Bobylev 2022; Chen et al. 2023). We have also presented the distributions of radio stars (see Section 3.1.4), OH/IR stars (see Section 3.1.2), and ICRF-defining sources in Figure 12, hoping to contribute to future links between GCRF and ICRF using SKA-VLBI. Makarov et al. (2019) used radio-loud AGNs to link GCRF and ICRF and found that the Gaia DR2 parallax zero-point is color dependent (for the effect, see also Gaia Collaboration et al. 2021; Lindegren et al. 2021), suggesting an uncorrected instrumental calibration effect. Linking GCRF and ICRF helps to identify systematic errors in Gaia astrometry which can be corrected in future releases (Rioja & Dodson 2020).

It is possible to determine Earth’s rotation and wobbles due to the influence of planets/other objects in the solar system as it orbits the Sun, by measuring the proper motion and/or parallax of Earth relative to an inertial coordinate system (e.g., ICRF; Fomalont & Reid 2004). The proper motion and parallax can also be used to investigate the motion of Earth’s crust and its subsequent effect on ground-based telescopes, as well as other important information relevant for geophysical research.

The precision of the ICRF is important for improving the tracking precision of spacecraft, beacons, and satellite orbits and for determining planetary ephemerides. Planetary ephemerides are an important tool, which can be used to calculate the positions of celestial bodies in the solar system at any given time, be incorporated into VLBI data correlators, used for navigation, and employed to predict and/or backtrack astronomical phenomena (e.g., Li 2008). Determination of modern-day planetary ephemerides requires improving the precision of monitoring planets and their orbiting (artificial) satellites (e.g., Folkner et al. 2014; Pitjeva & Pitjev 2018; Fienga et al. 2020; Park et al. 2021). Linking planets to the ICRF through VLBI orbital monitoring of artificial satellites can also effectively improve their pointing uncertainties (e.g., Folkner & Border 2015; Park et al. 2021).

The celestial bodies in the solar system, which define the barycentric celestial reference system (BCRS) and act as gravitational sources, can play an important role in GR tests (for a review see chapter 8 of Ni 2017). There are several ways to conduct a solar system test of GR:

(1)

Measure light deflection, the Shapiro time delay, and place constraints on the main parameter of parameterized post-Newtonian (PPN) formalism, $\gamma$.

(2)

Measure relativistic perihelion advance and the solar quadrupole moment.

(3)

Conduct ephemerides fitting.

(4)

Measure time variability of the gravitational constant and mass loss from the Sun.

(5)

Quantify frame-dragging effects.

(6)

Conduct lunar laser ranging tests of relativistic gravity.

SKA-VLBI will play an important role in at least points (1)–(5) above. For points (1) and (2), celestial bodies within the solar system act as gravitational sources. The role of SKA-VLBI in determining planetary ephemerides can be found in Section 3.3.3, while point (4) is often studied in conjunction with ephemerides fitting. SKA-VLBI can evaluate frame-dragging effects by measuring the proper motions of guide stars (for more details see the satellite mission Gravity Probe B (GP-B); Ratner et al. 2012; Shapiro et al. 2012, see also below).

Light deflection and the Shapiro time delay. The precision of $\gamma$ obtained by measurements of light deflection and the Shapiro time delay has rapidly improved in recent years (for more details see the reviews by Will 2015; Thompson et al. 2017). VLBI astrometry is an important technique that can lead to improved precision of $\gamma$. The precision of $\gamma$ obtained using the VLBI technique has reached $\sim 9\times 10^{-5}$ (Titov et al. 2018) where the Sun acts as a lens. Planets, such as Jupiter, have also been used to test GR using VLBI (e.g., Fomalont & Kopeikin 2003, 2008; Li et al. 2022c) and data from Gaia EDR3/DR3 (see Abbas 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), while light deflection by solar system objects will pose a nonnegligible restraining factor in astrometry by SKA-VLBI, affecting positional accuracies at the level $\lesssim$1 $\mu$as (Li et al. 2022b).

Taking the solar system planets as gravitational lenses, assuming a positional precision of 2 $\mu$as and there are $\sim$250 sources within 2^∘ (see Figure 6 at 15 GHz, where the FoV is set to be $\sim$2.1^′), and that light is deflected by $\sim$2 mas caused by major planets (e.g., Jupiter), it is expected that 10 epochs of observations will produce a measurement precision for $\gamma$ $\sim$10^-6. If the Sun acts as a lens, the deflection angle of light 1^∘ away from the Sun (which is favorable for successful observations; see Titov et al. 2018) can reach $\sim$500 mas, and the number of sources within 1^∘ is $\sim$63. Therefore, the measurement precision of $\gamma$ with 10 epochs of observations is expected to reach $\sim$$10^{-8}$, reaching the advanced level of the next few decades (see chapter 8 in Ni 2017). In such a case, using measurements of light deflection caused by the gravitational fields of celestial bodies in the solar system via SKA-VLBI may allow astronomers to test the (noninertial) motion and multipole moment effect of celestial bodies, and test and develop second-order or higher PPN formalisms and/or different metric theories of gravity (e.g., Damour & Nordtvedt 1993; Kopeikin & Makarov 2007; Fomalont et al. 2009; Li et al. 2022c). More interestingly, measurements of light deflection may be able to test for the presence of dark energy in the solar system, which requires astrometric precision of $\lesssim$ 10 $\mu$as (see Zhang 2022).

Relativistic perihelion advance. Another important component of solar system tests of GR is measuring the relativistic perihelion precession. This can be achieved by monitoring the orbits of planets or satellites in the solar system and measuring their annual precession. For example, for Mercury, the contribution of relativistic precession, $\Delta\phi$, is approximately 400 mas yr^-1 (see page 1113 in Misner et al. 1973; Laskar & Gastineau 2009; Mogavero & Laskar 2021; Brown & Rein 2023). If the positional precision of SKA-VLBI will be 2 $\mu$as, the measurement precision of $\Delta\phi$ with 1 yr of monitoring can reach $\sim 5\times 10^{-6}$. The relationship between $\Delta\phi$ and PPN parameters $\gamma$ and $\beta$ can be expressed by the following equation (see page 1116 in Misner et al. 1973)

where $G$ is the gravitational constant, $M_{\odot}$ and $R_{\odot}$ are the solar mass and radius, respectively, $a$ and $e$ are the semimajor axis and the eccentricity of the orbit of Mercury, respectively, $c$ is the speed of light, and $J_{2}$, the solar quadrupole moment, is $\sim 10^{-7}$ (see chapter 8 in Ni 2017). The second term is smaller than the first term by a factor of $\sim 6\times 10^{-4}$, and the precision of $J_{2}$ can currently reach $\sim$10^-2 (e.g., Pitjeva et al. 2022). Therefore, assuming an precision of $\gamma$ of $10^{-6}$, it is expected that SKA-VLBI will allow astronomers to test the PPN parameter $\beta$ at an precision of $\sim$10^-6. This precision will be comparable to the results obtained by fitting of Ephemerides of Planets and the Moon (EPM; e.g., 3$\times$10^-5 in Pitjeva 2013), allowing for mutual verification.

Ephemerides fitting and the time derivative of gravitational constant. As mentioned in Section 3.3.3, SKA-VLBI will likely play an important role in improving the precision of ICRF and planetary ephemerides. In fact, while fitting ephemerides, combined with orbital data of (artificial) satellites, GR can also be tested (such as the PPN parameters $\gamma$ and $\beta$ and the solar quadrupole moment, $J_{2}$); it may even be possible to obtain the time derivative of Newton’s gravitational constant (e.g., Anderson et al. 2002; Fienga et al. 2011; Pitjeva 2013; Verma et al. 2014) and whether dark matter present in the solar system (e.g., Pitjev & Pitjeva 2013; Pitjeva & Pitjev 2013); for more details see the review in chapter 8 in Ni (2017).

Frame-dragging effects. GP-B was a satellite mission that tested GR frame dragging, i.e., the Lense-Thirring effect, caused by the spin of Earth (see Ratner et al. 2012; Shapiro et al. 2012; Reid & Honma 2014). The task of VLBI in this research was to determine the proper motion of the guide star of the GP-B mission, the RS CVn binary IM Peg, i.e., $-$20.833 $\pm$ 0.090 mas yr^-1 and $-$27.267 $\pm$ 0.095 mas yr^-1 in R.A. and Decl. (J2000), respectively. As mentioned previously for IM Peg in Section 3.2.1, SKA-VLBI will be capable of measuring its proper motion with an precision of $\sim$4 $\mu$as yr^-1, thus improving the precision with which GR frame dragging can be tested. Moreover, the theoretically predicted Lense-Thirring effect signatures of the Galilean moons (such as Io, Europa, Ganymede, and Callisto) can, in principle, currently be detected with an astrometric precision of $\sim$10 mas (Iorio 2023). The flux densities of these satellites were over $\sim$10 mJy, as observed with the VLA at 15 GHz when Jupiter was $\sim$4.46 AU from Earth (de Pater et al. 1984). These flux densities are much higher than the sensitivity, $\Delta S$, listed in Table 3 (i.e., 6.9 $\mu$Jy beam^-1), indicating SKA-VLBI may be a suitable instrument to measure the Lense-Thirring effect on the Galilean moons of Jupiter.

HTBPSRs can be used to test GR (see Section 3.2.1). The measurement of HTBPSR orbital decay depends heavily on the fundamental parameters of the MW (e.g., $R_{0}$ and $\Theta_{0}$, see Section 3.3.1), where the acceleration of the Sun and pulsar in their Galactic orbits need to be considered (for the Sun, $\Theta^{2}_{0}/R_{0}$, and for a pulsar, $\Theta^{2}(R)/R$; for more details see Reid & Honma 2014; Deller et al. 2018; Reid et al. 2019b). This orbital decay is caused by gravitational radiation, as predicted by GR (Damour & Taylor 1991); thus measuring such decay and deducting the impact of the acceleration of the Sun and pulsar in Galactic orbit provide further tests of GR. A series of studies of the classical HTBPSR, B1913$+$16, showed that VLBA observations of HTBPSRs can test GR with an precision of $\sim 10^{-3}$ (Deller et al. 2018), but more HTBPSRs and more observations are required (see Section 3.2.1). If we keep the parameters in Deller et al. (2018) unchanged and simply replace their precision in parallax and proper motion by 6 $\mu$as and 19 $\mu$as yr^-1 at L2 band, respectively, the calculated “Galactic” correction caused by kinematic effects, $\dot{P}^{\mathrm{gal}}_{\mathrm{b}}$, to the observed orbital period derivative is $\dot{P}^{\mathrm{gal}}_{\mathrm{b}}=-(0.008\pm 0.001)\times 10^{-12}$ s s^-1, and the ROPR would be 1.0053 $\pm$ 0.0006, as determined by monitoring PSR B1913$+$16. The uncertainties of $\dot{P}^{\mathrm{gal}}_{\mathrm{b}}$ and ROPR are about seven and four times lower than those reported in Deller et al. (2018), respectively. It is also expected that more precise values of $R_{0}$ and $\Theta_{0}$ will result from higher-precision measurements of the parallaxes and proper motions of densely distributed HMSFRs in the MW (see Section 3.3.1), which may also lead to a more precise measurement of $\dot{P}^{\mathrm{gal}}_{\mathrm{b}}$. Therefore, SKA-VLBI astrometry will be able to test GR with high precision by observing HTBPSRs. GR has also been tested by monitoring the orbital decay of pulsar binaries (e.g., composed of a pulsar and a white dwarf, Guo et al. 2021) and double neutron star binaries (e.g., Ding et al. 2021).

Monitoring pulsars orbit around Sgr A^∗ at the GC can also allow astronomers to measure and test GR effects with high-precision astrometry (see Section 3.2.3; Reid & Honma 2014). GR can also be tested by measuring the proper motions of galaxies. The deflection of light from QSOs by a primordial gravitational wave background spanning $10^{-18}$–$10^{-8}$ Hz may also be measured or constrained using SKA-VLBI (see Gwinn et al. 1997; Book & Flanagan 2011; Truebenbach & Darling 2017).

Gravitational lenses, a variant of Einstein’s light deflection, are an important tool in astronomical research (Will 2015). According to the shapes of images, gravitational lensing can be divided into weak and strong, where the former causes images to be weakly deformed, and the latter results in multiple images (see chapter 8 of Aharonian et al. 2013). Gravitational microlensing can be thought of as a version of strong gravitational lensing, where the image separation is too small to be resolved (see part 4 in Schneider et al. 2006).

The application of strong gravitational lensing includes natural telescope that magnifies galaxies or galaxy clusters, enhances the resolution of the instruments, and is beneficial for observing high redshift sources; studies of cosmology; reconstruction of the mass distribution of the lens; etc. Multiple images have the same intrinsic spectrum, structure, and polarization properties. The observed differences between images (such as spectrum and polarization) can be used to study different propagation effects, such as dust extinction and reddening, free-free absorption, and Faraday rotation in the radio regime (for more details see chapter 8 of Aharonian et al. 2013). Such properties can also be used to measure the Hubble constant (e.g., the series of works by the TDCOSMO collaboration; see Millon et al. 2020). Gravitational microlensing allows astronomers to study dark energy (see part 4 in Schneider et al. 2006), although its prominent application is to search for exoplanets, dwarfs, and dark stars (such as neutron stars, white dwarfs, stellar black holes, etc.; Mao & Paczynski 1991; Zhu & Dong 2021; Kaczmarek et al. 2022). Weak gravitational lensing allows astronomers to reconstruct the mass profile of the lensing galaxy or galaxy cluster, study the mass-luminosity relationship, trace the universal density profile, and study the geometric configuration of the universe (for more details see chapter 8 of Aharonian et al. 2013).

The advantage of radio VLBI observations of the lensing is that they can provide information at all scales from Mpc to pc, allowing astronomers to study the mass distribution of lenses. However, to date there have been few gravitational lenses detected in radio bands (see chapter 8 of Aharonian et al. 2013). Hartley et al. (2021) found signs of the existence of strong- or microlensing in radio images, which helped them study possible radio-quiet QSOs. Radio studies partially assisted by theoretical expectations have presented some possible hints of microlensing, but without clear detections (e.g., Lecavelier des Etangs et al. 2013; Vedantham et al. 2020). Radio research on weak lensing is also beginning to prosper, in the form of several current surveys (e.g., using VLA and/or e-MERLIN) and forthcoming surveys (using SKA; e.g., Chang et al. 2004; Harrison et al. 2016, 2020). However, it is still challenging to identify weak gravitational lensing events using current radio equipment such as VLA $+$ e-MERLIN (e.g., Harrison et al. 2020), and there are not many resolved detections (Connor et al. 2022).

Koopmans et al. (2004) estimated that SKA will be able to detect $10^{6}$ strong gravitational lensing events in a half-sky general-purpose survey, of which $10^{5}$ could be easily detected. McKean et al. (2015) detailed the observing conditions, i.e., a $\theta_{\mathrm{beam}}$ of 0.25^′′–0.50^′′ and a depth of 3 $\mu$Jy beam^-1. For Galactic black hole microlensing events, it may be possible to achieve a rate of $\sim$10 yr^-1 using VLBA with a flux limit of 3 $\mu$Jy beam^-1 (see Karami et al. 2016). For exoplanetary microlensing, phase I of SKA-Low may reach a rate of $\sim$0.3 yr^-1 or better (see Shiohira et al. 2020). The high-sensitivity SKA will allow for the measurement of weak gravitational lenses in the radio at a level comparable to optical surveys, suggesting its useful application in future weak lensing surveys (see, e.g., chapter 8 of Aharonian et al. 2013; Rivi et al. 2016; Connor et al. 2022). For example, SKA phase I may be comparable to optical stage III experiments such as the Dark Energy Survey (Bonaldi et al. 2016; Harrison et al. 2016; Camera et al. 2017). VLBI with SKA-Mid will also be competitive in cosmology studies, especially for $\Lambda$CDM cosmology, where auto-correlations and/or cross-correlation with galaxy weak-lensing and Lyman-$\alpha$ forest observations can be employed (e.g., Dash et al. 2023). SKA-VLBI will have a higher resolution compared to SKA, making it suitable for strengthening SKA surveys, and which has great potential to bring about a major breakthrough in gravitational lensing research.

It is worth mentioning that since light deflection caused by a gravitational field is independent of wavelength, gravitational lensing is generally achromatic. However, chromatic microlensing may also be important, e.g., the presence of important chromatic microlensing in strong lensed quasars (Kayser et al. 1986). One of the important origins of chromatic microlensing may be plasma, while others may be chromatic refractive deflection or selective absorption (for more details see Sun et al. 2023). Such phenomena, especially plasma lensing, have been studied in the radio band (e.g., Draine 1998; Bisnovatyi-Kogan & Tsupko 2017; Tsupko & Bisnovatyi-Kogan 2020), and will therefore be one of the potential subjects of SKA-VLBI.