How often are radio telescopes used to measure parallax? When was this first done?

Question

The abstract of A magnetar parallax (also in MNRAS) says:

TE J1810-197 (J1810) was the first magnetar identified to emit radio pulses, and has been extensively studied during a radio-bright phase in 2003−2008. It is estimated to be relatively nearby compared to other Galactic magnetars, and provides a useful prototype for the physics of high magnetic fields, magnetar velocities, and the plausible connection to extragalactic fast radio bursts. Upon the re-brightening of the magnetar at radio wavelengths in late 2018, we resumed an astrometric campaign on J1810 with the Very Long Baseline Array, and sampled 14 new positions of J1810 over 1.3 years. The phase calibration for the new observations was performed with two phase calibrators that are quasi-colinear on the sky with J1810, enabling substantial improvement of the resultant astrometric precision. Combining our new observations with two archival observations from 2006, we have refined the proper motion and reference position of the magnetar and have measured its annual geometric parallax, the first such measurement for a magnetar. The parallax of 0.40±0.05 mas corresponds to a most probable distance 2.5+0.4−0.3 kpc for J1810. Our new astrometric results confirm an unremarkable transverse peculiar velocity of ≈200km s−1 for J1810, which is only at the average level among the pulsar population. The magnetar proper motion vector points back to the central region of a supernova remnant (SNR) at a compatible distance at ≈70 kyr ago, but a direct association is disfavored by the estimated SNR age of ~3 kyr.

This reports the first radio astrometric determination of parallax "for a magnetar".

Question(s):

1. How often are radio telescopes used to measure parallax? Is it done semi-regularly, or is this a rarity, used only in special cases?
2. When was the first time that the parallax of a distant object (e.g. outside of our solar system) measured for the first time using radio astrometric techniques?

Unfortunately "first time" is ambiguous, it could mean first time ever for this kind of measurement, or first time for a given object i.e. nobody knew the parallax of that particular object until the radio astrometric value was determined. It may be too much work to try to answer both. If that's the case, just indicate which kind of "first" is being reported.

Answer 1

I can partially answer this from a pulsar perspective.

Across the board, parallax measurements are a fairly new development in radio astronomy. Interferometers are required to attain any reasonable degree in accuracy, and so parallax measurements have come into play mainly in the last two decades, with instruments like the Very Long Baseline Array taking center stage. That said, we've had pulsar parallaxes for about 40 years. Salter, Lyne and Anderson (1979) determined parallaxes for six pulsars, although the measured parallax for B1929+10 was anomalously high and was quickly disputed (Backer & Sramek 1982). As of the turn of the century, only about a dozen pulsars had radio parallaxes (Toscano et al. 1999); that number has since increased significantly, although I don't know the current count.

Parallax has a complicated relationship with pulsars, because arguably we don't need it most of the time for the sake of intrinsic distance measurements. Radio waves are strongly affected by dispersion in the interstellar medium, as interactions between radio waves and free electrons delay the time of a signal's arrival by a frequency-dependent amount (it scales like $\nu^{-2}$). The magnitude of this scaling is determined by the dispersion measure, which is given by the line integral of the free electron number density over the path between the observer and the pulsar: $$\text{DM}=\int n_e\text{d}l$$ When observing a source, you can search over a variety of possible DMs and find the one that best removes this dispersion from the signal. Once you have the right DM, you can compare it to models of the Galactic electron number density (e.g. the NE2001 model) and use the source's right ascension and declination to determine how far away it is. Therefore, you can get a reasonable distance estimate in something like 15 minutes with a good single-dish telescope, rather than waiting months to perform measurements with an interferometer. Since the $\nu^{-2}$ drop-off means dispersion is irrelevant at other sections of the electromagnetic spectrum, radio astronomers have an extra tool in their toolboxes to determine distances.

(On the other hand, those models of electron number density had to be derived somehow - folks needed to know a priori distances to the pulsars in order to generate it. Parallax is one way to do this; alternatively, you could figure out if any of the pulsars being used for calibration belong to associations with known distances. Plus, sometimes the models are incomplete or wrong! The FAST GPPS survey (Han et al. 2021) turned up 11 pulsars with DMs larger than the maximum DM predicted by one or both of the NE2001 and YWM16 models, meaning that some sort of overdensities - say, more HII regions - need to be accounted for.)

You can also determine parallaxes without explicitly measuring them. Many pulsars and magnetars are studied using pulsar timing, which looks at when sets of pulses arrive and compare those to model times of arrival. The most commonly fit parameters are right ascension, declination, spin frequency $f$ (or period $P$), and its time derivative $\dot{f}$ (or period derivative $\dot{P}$), but for sources that aren't awful and for which you have enough of epochs of observations, you should be able to fit other quantities, including binary parameters for pulsars with a companion as well as proper motion and, yes, parallax in certain cases. If you can get enough observations on a large single-dish telescope, you could determine the parallax of a bright and stable source like a nice millisecond pulsar, albeit not to the $\sim10\;\mu\text{as}$ accuracy of a long-baseline interferometer like the VLBA.

When performing high-precision timing, we do have to account for how the Earth moves relative to the Solar System barycenter, leading to something called the Römer delay (times of arrival do get converted from the topocentric arrival time at an observatory to the arrival time at the SSB). This does lead to variations in arrival time on the order of $$\Delta_{R\odot}^{\text{max}}=\frac{1\;\text{AU}}{c}\cos\beta\approx500\cos\beta\;\text{seconds}$$ with $\beta$ the ecliptic latitude. This makes it difficult to fit for position near the ecliptic, which is where interferometric observations can come in handy for a timing model.

All that said, if you already have astrometry and know the proper motion and parallax of a source, your timing model might be significantly better, as the magnetar paper you link to points out. If you're really interested in high-precision timing, it might be worth going to the extra trouble. I'm certainly curious as to how more astrometry studies of magnetars will improve timing models.