Annu. Rev. Astron. Astrophys. 2014. 54: Copyright © 2014 by Annual Reviews. All rights reserved

2. ASTROPHYSICAL APPLICATIONS

Since distance is fundamental to astrophysical understanding, it should not be surprising that radio astrometry and parallax measurements are critical for characterizing a wide variety of phenomena and classes of sources. Here we briefly discuss some of the more important astrophysical applications of radio astrometry.

2.1. Galactic Structure

Surprisingly, we know little of the spiral structure of the Milky Way; there is considerable debate over the number of spiral arms, the nature of the central bar, and the values of the fundamental parameters R₀ (distance to Galactic center) and Θ₀ (circular rotation speed of the LSR). Major projects are underway to map the Milky Way with the VLBI Exploration of Radio Astrometry (VERA) array and the Very Long Baseline Array (VLBA) array, though the Bar and Spiral Structure Legacy (BeSSeL) survey. With typical parallax accuracy of ± 20 μas, and best accuracy of ± 5 μas, one can measure distances of 5 and 20 kpc, respectively, with ± 10% accuracy. For example, parallax data for the most distant source measured to date, the massive star forming region W 49, are shown in Figure 1. Over 100 distances to high-mass star forming regions have been measured using the astronomical "gold standard" technique of trigonometric parallax (see Reid et al. 2009b, Honma et al. 2012, Reid et al. 2014 and references therein).

Figure 1. Parallax data for W 49N measured with the VLBA after Zhang et al. (2013). Plotted are eastward position offsets versus time for H2O maser spots at an LSR velocity of 8.3 km s-1 relative to the background quasar J1905+0952 (shown in red) and a maser spot at an LSR velocity of 4.9 km s-1 relative to the background quasar J1922+0841 (shown in blue). The best fit proper motion has been removed, allowing the data to be overlaid and effects of parallax to be more clearly seen. This source has a parallax of 0.090 ± 0.006 mas, corresponding to a distance of 11.1 ± 0.8 kpc.

Attempts to decode the nature of the spiral structure of the Milky Way have long relied on kinematic distance estimates (by comparing the Doppler velocity of a source and that expected from a model of Galactic rotation with distance as a free parameter). One of the first high-precision VLBA parallaxes was for the massive star forming region W3(OH); Xu et al. (2006) found a distance of 1.95 ± 0.04 kpc, confirmed by Hachisuka et al. (2006). This distance was more than a factor of two less than its kinematic distance, demonstrating how unreliable kinematic distances can be. Now, with "gold standard" trigonometric parallaxes, the major spiral features of the Milky Way are, for the first time, being accurately located and spiral arm pitch angles measured (see Figure 2). Recent surprises include that the Local (Orion) arm, thought to be a minor structure, is longer and has far more on-going star formation than previously thought and rivals the Perseus spiral arm in the second and third Galactic quadrants (Xu et al. 2013).

With measured source coordinates, distance, proper motion, and radial velocity, one has full phase-space (3-dimensional position and velocity) information. These data can be modeled to yield an estimate of R₀, Θ₀, and the slope of the Galaxy's rotation curve. The latest modeling indicates that R₀ = 8.35 ± 0.16 kpc and Θ₀ = 251 ± 8 km s^-1 (Reid et al. 2014). Also, the rotation curve between 4 and 13 kpc from the Galactic center is very flat (Honma et al. 2007, Reid et al. 2014). Such a large value for Θ₀ (about 15% larger than the IAU recommended value of 220 km s^-1), if confirmed by other measurements, would have widespread impact in astrophysics, including increasing the total mass (including dark matter halo) of the Milky Way by about 50%, revising Local Group dynamics by changing the Milky Way's mass and velocities of Group members (when transforming from Heliocentric to Galactocentric coordinate systems), and increasing the expected signal from dark matter annihilation radiation.

Figure 2. Plan view of Milky Way. The background is an artist conception, guided by VLBI astrometry and Spitzer Space Telescope photometry (R. Hurt: IPAC). Dots are the locations of newly formed OB-type stars determined from trigonometric parallaxes using associated maser emission. The parallaxes were determined with the VLBA, VERA, and the EVN. Assignment to spiral arms (indicated by dot color) has been made by comparison to large scale emission of carbon monoxide in Galactic longitude–Doppler velocity space, independent of distance. The Galactic center is denoted by the plus (+) sign at (0,0) kpc; the Sun is labeled in yellow at (0,8.4) kpc. On this view, the Milky Way rotates clockwise.

2.2. Star Formation

Gould's Belt, a flattened structure of star forming regions of radius ≈ 1 kpc and centered ≈ 0.1 kpc from the Sun (toward the Galactic anticenter), contains most of the sites of current star formation near the Sun (e.g., the Ophiucus, Lupus, Taurus, Orion, Aquila Rift and Serpens star forming regions). Most of our knowledge about the formation of stars like the Sun comes from in-depth studies of these regions, for example, from the Spitzer c2d survey (Evans et al. 2009), the XMM Newton Extended Survey of Taurus (Güdel et al. 2008) and the Herschel Gould Belt Survey (André et al. 2010). Of course, knowledge of distance is critical for quantitative measures of cloud and young stellar object (YSO) sizes, masses, luminosities and ages. For such deeply embedded sources, which are optically invisible, typically distances had not been estimated to an accuracy of better than ± 30%. Since some physical parameters depend on distance squared or cubed, these parameters can be in error by factors of ≈ 2.

Trigonometic parallax measurements with VLBI techniques can be made by observing gyro-synchrotron emission from T Tauri objects, which is usually confined to a region of a few stellar radii, or from H₂O maser emission often associated with Herbig-Haro outflows. Currently, about a dozen YSOs parallaxes have been obtained, with up to 200 planned in this decade.

In the Ophiucus cloud, Loinard et al. (2008), using the VLBA, found that sources S1 and DoAr21 are at a mean distance of 120.0 ± 4.5 pc; Imai et al. (2007) using VERA found a consistent distance of 178_-37⁺¹⁸ pc, albeit with larger uncertainty, for the H₂O masers toward IRAS 16293-2422. The Taurus molecular cloud has been well studied and parallaxes for five YSOs have been reported from VLBA observations in a series of papers (Loinard et al. 2005, Loinard et al. 2007, Torres et al. 2007, Torres et al. 2009, Torres et al. 2012). These locate three YSOs associated with the L 1495 dark cloud at a distance of 131.4 ± 1.4 pc and, interestingly, T Tauri Sb at 146.7 ± 0.6 pc and HP Tau/G3 at 161.9 ± 0.9 pc. Clearly, these observations are tracing the 3-dimensional structure of the cloud, which is extended over about 30 pc along the line of sight, consistent with its extent on the sky of 10° (≈ 25 pc).

VERA observations have yielded parallaxes to YSOs in the Perseus molecular cloud. Observing H₂O masers, Hirota et al. (2008) measured parallax distances of 235 ± 18 pc for the SVS 13 toward NGC 1333 and 232 ± 18 pc for L 1448 C (Hirota et al. 2011). These measurements clarified the distance to the Perseus cloud, which was highly uncertain – estimated at between 220 pc (Cernis 1990) and 350 pc (Herbig & Jones 1983). The Serpens cloud provides another example where previous optical-based distance estimates varied considerably from 250 to 700 pc, with recent convergence toward the low end at ~ 230 ± 20 pc, as summarized by Eiroa, Djupvik & Casali (2008). However, Dzib et al. (2010) used the VLBA to obtain a trigonometric parallax distance for EC 59 of 414.9 ± 4.4 pc and suggested that faulty identification of dusty clouds in the foreground Aquila Rift might account for optical distance estimate.

The Orion Nebula is perhaps the most widely studied region of star formation. Prior to 1981, distance estimates for the Orion Nebula ranged from about 380 to 520 pc as summarized by Genzel et al. (1981). By comparing radial and proper motions (measured with VLBI observations) of H₂O masers toward the Kleinman–Low (KL) Nebula, an active region of star formation within the Orion Nebula, Genzel et al. estimated the distance to be 480 ± 80 pc. Since then, four independent trigonometric parallax measurements have been performed. Sandstrom et al. (2007) observed gyro-synchrotron emission from a YSO over nearly 2 yr and obtained a parallax distance of 389 ± 24 pc using the VLBA. Hirota et al. (2007) using VERA measured H₂O masers and determined a distance of 437 ± 19 pc for the KL region. Recently, two very accurate parallaxes have been published: Menten et al. (2007) used the VLBA and observed continuum emission from three YSOs and determined a distance of 414 ± 7 pc; Kim et al. (2008) using VERA and observing SiO masers estimated a distance of 418 ± 6 pc. The latter two measurements from different groups, using different VLBI arrays, different target sources, and different correlators and software are in excellent agreement. Together they indicate that the KL nebula/Trapezium region of the Orion Nebula is at a distance of 416 ± 5pc, nearly a 1% accurate distance! Since, the Orion Nebula is the subject of large surveys, from x-rays to radio waves, having such a "gold-standard" distance will enable precise estimates of sizes, luminosities, masses, and ages.

A large number (> 100) of parallaxes have been measured with the VLBA and EVN arrays for maser sources in high mass star forming regions (Bartkiewicz et al. 2008, Brunthaler et al. 2009, Hachisuka et al. 2009, Mollenbrock, Claussen & Goss 2009, Sanna et al. 2009, Xu et al. 2009, Zhang et al. 2009, Rygl et al. 2010, Sato et al. 2010a, Moscadelli et al. 2011, Xu et al. 2011, Sanna et al. 2012, Immer et al. 2013) and by the VERA array (Sato et al. 2008, Sato et al. 2010b, Oh et al. 2010, Ando et al. 2011, Honma et al. 2011, Kurayama et al. 2011, Matsumoto et al. 2011, Motogi et al. 2011, Nagayama et al. 2011a, Nagayama et al. 2011b, Niinuma et al. 2011, Shiozaki et al. 2011, Sakai et al. 2012). As the focus of these observations has been to better understand Galactic structure, we discuss these in Section 2.1.

2.3. Asymptotic Giant Branch Stars

Asymptotic Giant Branch (AGB) stars are in late stages of stellar evolution and have large convective envelopes and high mass-loss rates. They often exhibit maser emission from circumstellar OH, H₂O and SiO molecules, providing good targets for radio astrometry with VLBI. Accurate distances to AGB stars are necessary to constrain their physical parameters, such as size and luminosity, and these are crucial to test theories of late stages of stellar evolution. And, of course, distances are needed to calibrate the period-luminosity (P-L) relation of Mira variables, which can be used as standard candles, e.g., Whitelock, Feast & Van Leeuwen (2008). Giant stars are not good astrometric targets at optical wavelengths and the best optical parallaxes have accuracies poorer than a few mas. However, parallaxes based on radio observations of circumstellar masers has demonstrated more than a factor of ten better parallax accuracy, allowing much better calibration of the Mira P-L relation.

Early astrometric observations with the VLBA for red-giant OH masers demonstrated the potential for parallax measurements, achieving between 0.3 and 2 mas uncertainties (Langevelde et al. 2000, Vlemmings et al. 2003). For example, parallax distances for S CrB (418_-18⁺²¹ pc) and U Her (266_-28⁺³² pc) by Vlemmings & Langevelde (2007) are considerably more accurate than those measured by the Hipparcos satellite. Astrometry of OH masers at the relatively low observing frequency of 1.6 GHz are limited by uncompensated ionospheric effects, but these effects can be minimized by observations near solar minimum and using in-beam calibrators very close to the targets.

The first VLBI parallax of an AGB star using H₂O masers at 22 GHz revealed a distance for UX Cyg, a long period Mira variable, of 1.85_-0.19^+0.25 kpc (Kurayama, Sasao & Kobayashi 2005). Since then the VERA array has been used to obtain parallaxes for many Mira and semi-regular variables, including S Crt (Nakagawa et al. 2008), SY Scl (Nyu et al. 2011), and RX Boo (Kamezaki et al. 2012). Of particular interest are the astrometric measurements for the symbiotic system R Aqr (Kamohara et al. 2010) and the parallax distance of 218_-11⁺¹² pc (Min et al. 2013). Future observations may enable one to trace binary's orbital motion.

Red supergiants are rare objects, typically at kpc distances. At these distances, and owing to their very large sizes and irregular photospheres, they are beyond the reach of optical parallax measurements. Choi et al. (2008) observed H₂O masers toward VY CMa, one of the best-studied red supergiants, and found a parallax distance of 1.14_-0.09^+0.11 kpc with VERA. This distance was later confirmed by Zhang et al. (2012b), who used the VLBA and found a distance of 1.20_-0.10^+0.13 kpc. Generally, the distance had been assumed to be 1.5 kpc, requiring an extraordinary luminosity of 5 × 10⁵ L_⊙; the parallax distances reduce the luminosity estimate by 40% to a more reasonable value. VLBI parallaxes accurate to ± 10% have been obtained for several other red supergiants, including S Per (Asaki et al. 2010), NML Cyg (Zhang et al. 2012b), and IRAS 22480+6002 (Imai et al. 2012).

Proto-Planetary Nebulae are in the final stage of evolution of intermediate-mass stars, linking the AGB and planetary nebula phases. These sources are known to exhibit bipolar outflows with a velocities exceeding ~ 100 km s^-1; they can have strong H₂O masers and are often called "water-fountains." Parallaxes have been measured for IRAS 19134+2131 (Imai, Sahai & Morris 2007], IRAS 19312+1950 (Imai et al. 2011), IRAS 18286-0959 (Imai et al. 2013), and K 3-35 (Tafoya et al. 2011).

2.4. X-ray Binaries

The first trigonometric parallax for a black hole candidate was for the X-ray binary V404 Cyg (Miller-Jones et al. 2009), indicating a distance of 2.39 ± 0.14 kpc. This value was significantly lower than previously estimated and indicated that its 1989 outburst was not super-Eddington. Its peculiar velocity, derived from the radio proper motion, is only about 40 km s^-1, suggesting that it did not receive a large natal "kick" from an asymmetric supernova explosion. This differs from many pulsars, which often have order of magnitude larger peculiar motions (see Section 2.5).

The long-standing uncertainty over the distance to Cyg X-1 (see, e.g., Caballero-Nieves et al. 2009) limited understanding of this famous binary, including whether or not the unseen companion was a black hole. Recently a trigonometric parallax measurement with the VLBA yielded a distance of 1.86 ± 0.12 kpc (Reid et al. 2011). Knowing the distance to the binary removed the mass–distance degeneracy that limited the modeling of optical/IR data (light and velocity curves) and revealed that the unseen companion in Cyg X-1 has a mass of 14.8 ± 1.0 M_⊙ (Orosz et al. 2011). This mass confidently exceeds the limit for a neutron star and firmly established it as a black hole. Once the masses of the two stars were accurately determined, X-ray data could be well modeled. This revealed that black hole spin is near maximal and, since the binary is too young for accretion to have appreciably spun up the black hole, most of the spin angular momentum is probably natal (Gou et al. 2011). Finally, the 3-dimensional space-motion of the binary, from the radio astrometry and optical Doppler shifts, confirms Cyg X-1 as a member of the Cyg OB3 association, and the lack of evidence for a supernova explosion in this region suggests that the black hole may have formed via prompt collapse without an explosion (Mirabel & Rodigues 2003).

Recently, using the EVN and the VLBA, Miller-Jones et al. (2013) obtained a parallax distance of 114 ± 2 pc for SS Cyg, a binary composed of a white and red dwarf. This distance is significantly less than an optical parallax measured with the Hubble Space Telescope of 159 ± 12 pc (Harrison et al. 1999). The larger optical distance required the source to be significantly more luminous and proved difficult to reconcile with accretion disk theory. However, the smaller distance from radio astrometry seems to resolve this problem.

2.5. Pulsars

Pulsar radio emissions are extremely compact and relatively bright, which make them suitable for VLBI observations, and radio astrometry of pulsars dates back to the 1980's. The first interferometric parallax measurements were by Gwinn et al. (1986), who reported parallaxes for two pulsars using a VLBI array which included the Arecibo 305-m telescope. Later Bailes et al. (1990a) measured a parallax for PSR 1451-68 of 2.2 ± 0.3 mas, using the Parkes–Tidbinbilla Interferometer in Australia, and determined a line-of-sight average interstellar electron density of 0.019 ± 0.003 cm^-3 by combining the distance and dispersion measures, suggesting that the interstellar medium in the Solar neighborhood is typical of that over larger scales in the Galaxy. Pulsar proper motions for 6 pulsars conducted with the Parkes–Tidbinbilla Interferometer revealed that motions away from the Galactic plane were between 70 and 600 km s^-1 (Bailes et al. 1990b). These early studies provided distances and confirmed the expectation that pulsars are high-velocity objects, most-likely due to "kicks" associated with their parent supernova explosions.

Most recent pulsar astrometry uses "in-beam" calibrators to greatly improve positional accuracy. Other improvements include gating the pulsar signal during correlation to improve signal-to-noise ratios and performing ionospheric corrections based on multi-frequency phase fitting. With these improvements and using the VLBA a parallax with better than ± 10% accuracy was obtained for PSR B0950+08 at a distance of 280 pc (Brisken et al. 2000), and nine other pulsar parallaxes with distances between 160 and 1400 pc have been reported (Brisken et al. 2002).

By observing at higher frequencies (5 GHz instead of 1.6 GHz), in order to reduce the effects of the ionosphere, higher accuracy pulsar parallaxes have been obtained, for example, for PSR B0355+54 (0.91 ± 0.16 mas) and PSR B1929+10 (2.77 ± 0.07 mas) (Chatterjee et al. 2004). More recently, Chatterjee et al. (2009) obtained results for 14 pulsars with the VLBA, including a parallax for the most distance pulsar yet measured: PSR B1514+09 with a parallax of 0.13 ± 0.02 mas, corresponding to a distance of 7.2_-1.2^+1.3 kpc. With this sample, it is clear that most pulsars are moving away from the Galactic plane with speeds of hundreds of km s^-1.

Astrometry also provides a unique opportunity to constrain pulsar birth places through velocity and distance measurements. For instance, Campbell et al. (1996) used a global VLBI array to measure the parallax and proper motion for PSR B2021+51, which ruled out the supernova remnant HB 21 as the origin of the pulsar, as it implied an improbable low age of only 700 years. Similar studies have been performed by others (Dodson et al. 2003, Ng et al. 2007, Chatterjee et al. 2009, Bietenholz et al. 2013).

Astrometric observations also provide information on the physical properties of pulsars. For example, Brisken et al. (2003) combined a VLBA parallax for PSR B0656+14 with a thermal X-ray emission model to constrain the stellar radius between 13 and 20 km. Deller et al. (2012b) measured a parallax for the transitional millisecond pulsar J1023+0038 with the VLBA. Their distance of 1368_-39⁺⁴² pc was twice that predicted by the standard interstellar plasma model. When combined with timing and optical observations of this binary system, the new distance indicated a mass of M ~ 1.71 ± 0.1 M_⊙, suggesting that it is a recycled pulsar. Deller et al. (2009b) conducted astrometric observations of seven pulsars in the southern hemisphere using Australian Long Baseline Array (LBA). Their new distance to PSR J0630-2834 required the efficiency of conversion of spin-down energy to X-rays to be less than 1%, an order of magnitude lower than previous estimates using a less accurate distance.

Magnetars, pulsars with extremely strong magnetic field, are also interesting astrometric targets. Helfand et al. (2007) observed XTE J1810-197 with the VLBA and obtained a proper motion of 212 km s^-1, suggesting that this magnetar has a lower space motion than theoretical predicted (Duncan & Thompson 1992). Recently, Deller et al. (2012a) found a similarly low space motion for the magnetar J1550-5418.

Finally, pulsar astrometry can be critical for testing the constancy of physical parameters. Using the VLBA, Deller et al. (2008) observed J0437-4715, a milli-second binary pulsar with a white dwarf companion. They obtained a very precise parallax distance of 156.3 ± 1.3 pc. Comparing this to a kinematic distance from pulsar timing gives strong limits on unmodeled accelerations, which provide a limit on the constancy of the Gravitational constant of /G = (-5 ± 26) × 10^-13 yr^-1. This constraint is consistent with those obtained from lunar laser ranging as well as gravitational wave backgrounds.

2.6. Radio Stars

The Algol system is an eclipsing binary (with a period of 2.9 days, consisting of B8 V primary and K0 IV secondary) and a distant companion with an orbital period of 1.86 yr. VLBI astrometry by Lestrade et al. (1993) revealed that the radio emission originates from the K0 subgiant and traced the orbital motion of eclipsing binary, indicating that the orbit of the binary is nearly orthogonal to that of the tertiary companion. Observations of another hierarchical triple (Algol-like) system, UX Ari, by Peterson et al. (2011) detected the acceleration of the tight binary caused by the tertiary star. This acceleration measurement dynamically constrains the mass of the tertiary to be ≈ 0.75 M_⊙, a value consistent with a spectroscopic identification of a K1 main-sequence star.

Astrometric observations of radio stars allow one to accurately tie the fundamental radio and the optical reference frames. A comparison between the radio and preliminary Hipparcos frames by Lestrade et al. (1995) revealed systematic discrepancies that could be removed by a global rotation (and its time derivative). Further VLBI observations by Lestrade et al. (1999) achieved parallax accuracies of ≈ 0.25 mas for nine sources, whose distances ranged from 20 to 150 pc.

Dzib et al. (2013) monitored radio emission from colliding winds in Cyg OB2#5 and obtained a marginal parallax of 0.61 ± 0.22 mas, consistent with other distance measurements of Cyg X regions (Rygl et al. 2012, Zhang et al. 2012b). These observations also revealed a high radio-brightness temperature ( 10⁷ K), providing information for modeling the stellar winds.

2.7. Star Clusters

The Hyades and Pleiades clusters play a pivotal role in quantitative astrophysics, serving as pillars of the astronomical distance ladder. Recently, the Hipparcos space mission, which measured ≈ 100,000 stellar parallaxes with typical accuracies of ± 1 mas, presented a (revised) parallax distance of 120.2 ± 1.9 pc for the Pleiades (van Leeuwen 2009). This result has been quite controversial, since a variety of other techniques, including main-sequence fitting, generally give distances between 131 and 135 pc. Using the HST fine guidance sensor, Soderblom et al. (2005) measured relative trigonometric parallaxes for three Pleiads, which, after correction to absolute parallaxes, average to a distance of 134.6 ± 3.1 pc.

In an attempt to provide a totally independent and straight-forward distance to the Pleiades, Melis et al. (2014) have used the VLBA with the Green Bank and Arecibo telescopes to measure absolute parallaxes to several Pleiads that display compact radio emission. Preliminary results suggest a cluster parallax near 138 pc, with an uncertainty of less than ± 2 pc (including measurement error and cluster depth effects). This result seems to rule out the Hipparcos value, and it may even be in some tension with the ensemble of astrophysical-based distance indicators. Since the source of error for the Hipparcos parallax for the Pleiades has not been convincingly established, there could be concern for the Gaia mission, which is targeting a parallax accuracy of ± 20 μas, since Gaia might inherit some unknown systematics from Hipparcos. Intercomparison of high accuracy VLBA parallaxes with those from Gaia will provide a critical cross checking.

2.8. Sgr A*

Sgr A*, the candidate super-massive black hole (SMBH) at the center of the Galaxy, is a strong radio source. It is precluded from optical view by > 20 mag of visual extinction but can sometimes be detected when flaring at 2.2 μm wavelength (through 3 mag of extinction at this wavelength). Astrometric observations in the infrared of stars orbiting an unseen mass have provided compelling evidence of a huge mass concentration, almost surely a black hole (Ghez et al. 2008, Gillessen et al. 2009). These observations require a grid of sources with accurate positions relative to Sgr A* to calibrate the infrared plate scale, rotation, and low-order distortion terms. The calibration sources have been provided by VLA and VLBA astrometric observations relative to the radio bright Sgr A* of SiO and H₂O masers in the circumstellar envelopes of red giant and supergiant stars within the central pc (Reid et al. 2003). This allowed location of the position of Sgr A* on infrared images, which matched that of the gravitational focal position of the stellar orbits to ≈ 1 mas accuracy, as well as confirming Sgr A*'s extremely low luminosity (Menten et al. 1997).

Astrometric observations of Sgr A* at 7 mm wavelength, relative to background quasars with the VLBA, yielded the angular motion of the Sun about the Galactic center and placed extremely stringent limits on the mass density of the SMBH candidate. While stars orbiting Sgr A* have been observed to move with speeds exceeding 5000 km s^-1 (Schödel et al. 2002), the velocity component perpendicular to the Galactic plane of Sgr A* is less than ≈ 1 km s^-1, requiring that most of the 4 × 10⁶ M_⊙ required by the stellar orbits is tied to the radiative source Sgr A* (Reid & Brunthaler 2004). Since the millimeter wavelength emission from Sgr A* is confined to a ~ 1 Schwarzschild radii region (Doeleman et al. 2008), the implied mass density is approaching that theoretically expected for a black hole (Reid 2009).

2.9. Megamasers and the Hubble Constant

The Hubble constant, H₀, is a critical cosmological parameter, not only for the extragalactic distance scale, but also for determining the flatness of the Universe and the nature of dark energy. Some active galactic nuclei (AGN) with thin, edge-on accretion disks surrounding their central super-massive black hole (SMBH) exhibit H₂O "megamaser" emission, with bright maser spots coming from clouds in Keplerian orbit about the SMBH. Astrometric observations can be used to map the positions and velocities of these clouds. Coupled with time monitoring of the maser spectra, which allows direct measurement of cloud accelerations (by tracking the velocity drift over time of maser features), one can obtain a geometric (angular-diameter) distance estimate for the galaxy.

Observations of the archetypal megamaser galaxy, NGC 4258, demonstrated the power of radio astrometric observations for better understanding of AGN accretion disks (Herrnstein et al. 2005) and yielded an accurate distance of 7.2 ± 0.5 Mpc to the galaxy (Herrnstein et al. 1999). Since NGC 4258 is nearby and its (unknown) peculiar motion is likely to be a large fraction of its cosmological recessional velocity (≈ 500 km s^-1), one cannot directly measure H₀ by dividing velocity by distance. However, since Cepheid variables have been observed in NGC 4258, one can use the radio distance to recalibrate the zero-point of the Cepheid period-luminosity relation (Macri et al. 2006) and then revise estimates of H₀ (Riess, Fliri & Valls-Gabaud 2012, Riess et al. 2012). Recently, Humphreys et al. (2013) analyzed a decade of observations of NGC 4258 and, via more detailed modeling of disk kinematics, obtained an extremely accurate distance of 7.60 ± 0.23 Mpc, which constrains H₀ = 72.0 ± 3.0 km s^-1 Mpc^-1.

By observing H₂O masers in galaxies like NGC 4258, but more distant and well into the "Hubble flow," the Megamaser Cosmology Project (MCP) is obtaining direct estimates of H₀. For galaxies in the Hubble flow, unknown galaxy peculiar motions are a small source of systematic uncertainty ( 5%). Results for the megamaser galaxy UGC 3789 (Reid et al. 2009c, Braatz et al. 2010) have yielded H₀ = 68.9 ± 7.1 km s^-1 Mpc^-1 (Reid et al. 2013) and for NGC 6264 H₀ = 68 ± 9 km s^-1 Mpc^-1 (Kuo et al. 2013). Based on these results, and preliminary results for Mrk 1419, Braatz et al. (2013) find a combined result of H₀ = 68.0 ± 4.8 km s^-1 Mpc^-1. The goal of the MCP is measurement of ≈ 10 megamaser galaxies, each with an accuracy near ± 10%, which should yield a combined estimate of H₀ with ± 3% accuracy. While, formally, a ± 3% uncertainty may be slightly larger than claimed by other techniques, the megamaser method is direct (not dependent on standard candles) and totally independent.

2.10. Extragalactic Proper Motions

In the concordance ΛCDM cosmological model, galaxies grow hierarchically by accreting smaller galaxies. Nearby examples of galaxy interactions are found in the environments of the Milky Way and the Andromeda Galaxy, the dominant galaxies in the Local Group. In the past, only radial velocities for Local Group galaxies were known and statistical approaches had to be used to model the system. While the radial velocity of Andromeda indicates that it is approaching the Milky Way, without knowledge of its proper motion one cannot know, for example, if the two galaxies are on a collision course or if they are in a relatively stable orbit.

Astrometric VLBI observations have yielded both the internal angular rotation and the absolute proper motion of M33, a satellite of Andromeda (Brunthaler et al. 2005). The angular rotation of M33 was the focus of the van Maanen–Hubble debate in the 1920's (van Maanen 1923, Hubble 1926), with van Maanen claiming an angular rotation of 20 ± 1 mas y^-1. Such a large angular rotation required it to be nearby and part of the Milky Way, in order to avoid implausibly large rotation speeds. Indeed, Shapley forwarded this as evidence that spiral nebulae were Galactic objects in the famous Shapley–Curtis debate. The angular rotation measured by Brunthaler et al. is ~ 1000 times smaller than van Maanen claimed, and, of course, consistent with an external galaxy (see Figure 3). Coupled with knowledge of the H I rotation speed and inclination of M33, the angular rotation rate yields a direct estimate of distance ("rotational parallax") of 730 ± 168 kpc, consistent with standard candle estimates. Significant improvement in the accuracy of this distance estimate can come from better H I data and a longer time baseline for the proper motion measurements.

Figure 3. Image of M33, a satellite of the Andromeda galaxy, with the locations and measured proper motions of H2O masers in two regions of massive star formation (Brunthaler et al. 2005). Both the relative motions (i.e., the van Maanen experiment) and the absolute motions with respect to a background quasar have been measured. The relative motions gives a "rotational parallax" distance, and the absolute motion can be used to constrain the mass of the Andromeda galaxy.

Absolute proper motions (with respect to background quasars) have been measured for two Andromeda satellites: M33 and IC 10. Assuming these galaxies are gravitationally bound to Andromeda, their 3-dimensional motions yield a lower mass limit for Andromeda of > 7.5 × 10¹¹ M_⊙ (Brunthaler et al. 2007). Of course, knowledge of the proper motion of Andromeda would refine this estimate and is key to unlocking the history and fate of the Local Group. Observations with the upgraded bandwidth of the VLBA and the Green Bank and Effelsberg 100-m telescopes are underway to measure the proper motion of M 31* (the weak AGN at the center of Andromeda) and the motions of a handful of H₂O masers recently discovered in the galaxy (Darling 2011). Awaiting a direct measurement of the absolute proper motion of Andromeda, one can use the fact that M33 has not been tidally disrupted by a close encounter with Andromeda in the past to place constraints on the motion of Andromeda. In this way Loeb et al. (2005) showed that Andromeda likely has a proper motion of ~ 100 km s^-1.

2.11. Tests of General Relativity

The dominant uncertainty in modeling the Hulse-Taylor binary pulsar and measuring the effects of gravitational radiation on the binary orbit comes from uncertainty in the accelerations of the Sun (Θ₀² / R₀) and the pulsar (Θ(R)² / R) as they orbit the center of the Galaxy (Damour & Taylor 1991). Using recently improved values for the fundamental Galactic parameters reduces the uncertainty in the binary orbital decay expected from gravitational radiation by nearly a factor of four compared to using the IAU recommended values (Reid et al. 2014). The dominant uncertainty in the general relativistic test parameter is now dominated by the uncertainty in the pulsar distance, and a VLBI parallax for the binary accurate to ± 14% would bring the contribution from distance uncertainty down to that of Galactic parameter uncertainty.

Deller, Bailes & Tingay (2009a) obtained an accurate parallax distance of 1.15_-0.16^+0.22 kpc for the pulsar binary J0737-3039 A/B. Given that these pulsars are only a kpc distant (not ~ 10 kpc as is the Hulse-Taylor pulsar), with this parallax accuracy uncertainties in the correction for the effects of Galactic accelerations are an order of magnitude smaller than for the Hulse-Taylor system. Thus, with perhaps another decade of pulsar timing, one might achieve a test of the effects of gravitational radiation predicted by general relativity at 0.01% level.

Pulsars orbiting the super-massive black hole at the Galactic center, Sgr A*, may show general relativistic effects that can be measured and tested based on high-accuracy astrometric observations. Pulsars near the Galactic center are strongly affected by interstellar scattering, which makes it difficult to observe these pulsars at 10 GHz. For instance, the recently-discovered transient magnetar PSR J1745-2900 (Mori et al. 2013), located only 3" from Sgr A*, is highly likely to be in the Galactic center region. If another pulsar is found closer to Sgr A*, it could prove to be an excellent target for testing general relativity with unprecedented accuracy.

Gravity Probe B (GP-B) was a satellite mission to test general relativistic frame-dragging (Lense–Thirring effect) caused by the spin of the Earth. The satellite was equipped with four ultra-stable gyroscopes, allowing satellite altitude variation to be measured with unprecedented accuracy. Based on the data from the four gyroscopes spanning ~ 1 year, Everitt et al. (2011) reported a geodetic drift at -6601.8 ± 18.3 mas y^-1 and a frame-dragging drift at -37.2 ± 7.2 mas y^-1. These results are fully consistent with the predictions of Einstein's theory of general relativity (-6606.1 mas y^-1 and -39.2 mas y^-1, respectively). VLBI astrometry played a fundamental role in the calibration of the GP-B data. IM Peg (HR 8703), an RS CVn type binary system, was used as the reference star. Its parallax and proper motion were accurately measured with VLBA observations spanning 5 years (Shapiro et al. 2012 and references therein). Based on the astrometry of IM Peg relative to 3C 454.3, the absolute proper motion of IM Peg was determined with an accuracy better than 0.1 mas y^-1 (Ratner et al. 2012), which provided the fundamental basis for measuring the geodetic and frame dragging effect by GP-B.

The classical test of general relativity, measuring gravitational bending of star light by the Sun (Dyson, Eddington & Davidson 1919), has been repeated with radio interferometry by observing background radio sources and has provided one of the most accurate tests of general relativity. The most recent analysis of VLBI data reported that the post-Newtonian parameter γ is consistent with unity to an accuracy of one part in 10^-4 (Lambert & Le Poncin-Lafitte 2009, Fomalont et al. 2009, Lambert & Le Poncin-Lafitte 2011). Additionally, Fomalont & Kopeikin (2003) detected the deflection of radio waves by Jupiter, including the retarded delay caused by Jupiter's motion, and claimed that this constrained the speed of gravitational waves, although this claim is controversial (see Asada 2002, Will 2003, Fomalont & Kopeikin 2009).

2.12. Extrasolar Planets and Brown Dwarfs

A radio astrometric search for extrasolar planets was conducted by Lestrade et al. (1996) toward σ² CrB over a 7.5 year period. After removing the effects of parallax and proper motion, their post-fit residuals of < 0.3 mas excluded a Jupiter-mass planet orbiting at a radius of 4 AU from the central star. Guraido et al. (1997) observed the active K0 star AB Dor with the Australian LBA and, combined with HIPPARCOS data, inferred a companion with a mass of ~ 0.1M_⊙. These studies demonstrate the potential of radio astrometry to detect low-mass companions, including brown dwarfs and planets.

Low-mass stars (< 1 M_⊙) could have habitable planets which are difficult to detect with optical radial-velocity techniques, because the stars are faint and activity can distort their line profiles, reducing the accuracy of velocity measurements. Thus, radio astrometric planet searches are complementary to other approaches. The Radio Interferometric Planet (RIPL) Search (Bower et al. 2009) and the Radio Interferometric Survey of Active Red Dwarfs (Gawronski, Gozdziewski & Katarzynski 2013) seek to detect the effects of planets around nearby low-mass stars, specifically active M dwarfs. For RIPL, the demonstrated astrometric accuracy is 260 μas, which for GJ 897A limits a planetary companion at 2 AU radius to < 0.15 M_J (Bower et al. 2011).

2.13. AGN Cores

Active Galactic Nuclei (AGNs) often show very strong and compact radio emission, with brightness temperatures up to and exceeding 10¹² K. Bartel et al. (1986) measured the relative positions of a radio bright pair of QSOs, NRAO 512 and 3C 345, with a global VLBI network, revealing that the core positions of the sources were stable to within ± 20 μas y^-1. This stability corresponds to an upper limit of ~ 1 c for the core-motions, which is significantly lower than observed in the jets of super-luminal sources. Marcaide, Elosegui & Shapiro (1994), Rioja et al. (1997), Guirado et al. (1995) and Marti-Vidal et al. (2008) reported little if any relative proper motion between other QSO pairs (at levels of ~ 10 μas y^-1), consistent with contamination of stationary cores with some expanding plasma. Such studies point out a source of limiting "noise" when using radio loud QSOs for establishing fundamental reference frames.

In addition to measuring relative positions between two QSOs, one can precisely measure positions of jet components within a source at different frequencies. In this manner, Hada et al. (2011) measured the theoretically predicted core-shift as a function of observing frequency (Blandford & Königl 1979) for the super-massive black hole of M87 (see 4) and located the black hole relative to the jet emission with an accuracy of ~ 10 R_sch. Kovalev et al. (2008) and Sokolovsky et al. (2011) have confirmed that frequency-dependent core-shifts, such as observed in M87, are common in large samples of AGNs with prominent jets. Bietenholz, Bartel & Rupen (2004) used SN 1993J in M81 as a position reference for multifrequency imaging of M 81* (the AGN in that galaxy) to accurately locate the super-massive black hole. Later Marti-Vidal et al. (2011) found that the M 81* core position appears variable at lower frequencies, and they suggest this is a result of jet precession.

Figure 4. The apparent shift in the core of M87 as a function of observing frequency after Hada et al. (2012). This effect can be explained by frequency dependent optical depths and allows the position of the supermassive black hole to be accurately located on the images.

Porcas (2009) considered the effects of an AGN core-shifts on group-delay astrometry, used for measuring antenna locations for geodetic VLBI and for establishing reference frames (e.g., the ICRF). Usually group-delay measurements are conducted at 8.4 GHz and, to remove propagation delays in the ionosphere, simultaneous observations are also made at 2.3 GHz. When removing the dispersive component of delay to correct for ionospheric propagation delay, generally one assumes that AGN core positions are the same at the two observing frequencies. However, the core-shift effect on dual-frequency geodetic VLBI observations can be comparable to a position error of ~ 170 μas and should be measured and compensated for in the future, for example, when tying the radio reference frame with a new optical frame constructed by Gaia (Bourda et al. 2011).

2.14. Satellite Tracking

The technique of high-accuracy VLBI astrometry can be applied to locating spacecraft, using telemetry signals as radio beacons. In fact, astrometric observations of radio beacons placed on the moon by Apollo missions were among the earliest applications of phase-referencing VLBI. The Apollo Lunar Surface Experiments Package and Lunar Roving Vehicle radio transmitters from various Apollo missions spanned hundreds of km across the Lunar surface. These transmitters were observed by Counselman et al. (1973) and Salzberg (1973), who measured their separations with an accuracy of ~ 1 m (~ 0.5 mas). Later Slade et al. (1977) was able to tie these transmitters to the celestial reference frame with a positional accuracy of ~ 1 mas. King, Counselman & Shapiro (1976) combined VLBI observations of Lunar transmitters with ranging data to better estimate the mass of the Earth-Moon system and the moment-of-inertia ratios of the moon. (See Lanyi, Bagri & Border 2007 for a recent review.)

Recently, high-accuracy radio astrometry played a part in generating a precise map of the Lunar gravity field in conjunction with the SELENE mission (Goossens et al. 2011). This was made possible by the high relative-position accuracy possible with in-beam astrometry (Kikuchi et al. 2009). This method was originally developed in 1990's for observing artificial radio sources near Venus (Border et al. 1992, Folkner et al. 1993) and made possible measurements of wind speed as a function of altitude, by monitoring the trajectories of probes released into the Venusian atmosphere (Counselman et al. 1979, Sagdeyev et al. 1992).

In the outer solar system, VLBI astrometry played a role in the PHOBOS-2 (Hildebrand et al. 1994) and Mars Odyssey missions (Antreasian et al. 2005). The motion of the Huygens probe as it descended in Titan's atmosphere revealed the vertical profile of its wind speed (Witasse et al. 2006). The Cassini satellite itself was used to trace the gravity field of Saturn, and Jacobson et al. (2006) combined VLBI data with optical observations and Doppler ranging to better constrain the mass and potential of the Saturnian system. Also, VLBA observations of Cassini have measured the center-of-mass of the Saturnian system with an accuracy of 2 km (0.3 mas) with respect to the ICRF (Jones et al. 2011). The potential of radio astrometry for improving future space missions has been reviewed by Duev et al. (2012).