1 INTRODUCTION
The past decade has ushered in an epoch of precision astrometry, with increases in sensitivity and enhanced processing techniques permitting very long baseline interferometers to make astrometric measurements accurate to a few tens of microarcseconds. This enables the measurement of model-independent parallax distances for radio-emitting objects out to several kiloparsecs, and proper motions for radio sources anywhere in the Galaxy (e.g. Brunthaler et al. Reference Brunthaler2011; Loinard et al. Reference Loinard, Mioduszewski, Torres, Dzib, Rodríguez, Boden, Henney and Torres-Peimbert2011), and, over a sufficiently long time baseline, out to Local Group objects (Brunthaler et al. Reference Brunthaler, Reid, Falcke, Greenhill and Henkel2005, Reference Brunthaler, Reid, Falcke, Henkel and Menten2007).
As Galactic objects with radio-emitting jets, X-ray binaries provide a potential set of astrometric targets that can be used to study jet physics and the formation of compact objects, and for which geometric parallax distances can be invaluable in constraining fundamental system parameters such as peak luminosity (relative to the Eddington luminosity) and black hole spin. However, the radio emission from X-ray binaries depends strongly on the X-ray spectral state (see, e.g. Fender, Belloni, & Gallo Reference Fender, Belloni and Gallo2004, for a review), and is not always suitable as an astrometric target. The radio emission at any particular wavelength is brightest at the peak of sporadic (and unpredictable) outbursts, and typically arises from relativistically-moving jet knots that are no longer causally connected to the binary system (e.g. Mirabel & Rodríguez Reference Mirabel and Rodríguez1994). The radio emission is then quenched by a factor of at least several hundred (Russell et al. Reference Russell, Miller-Jones, Maccarone, Yang, Fender and Lewis2011) during the soft, thermal-dominant X-ray state seen following the peak of the outburst. Except for the few outbursts without an associated ejection event (e.g. Rushton et al. Reference Rushton2012; Paragi et al. Reference Paragi2013), this implies that astrometric observations can only be carried out in the hard and quiescent states, when the faint, flat-spectrum radio emission is believed to arise from a relatively steady, compact jet that is causally connected to the binary system (e.g. Fender Reference Fender2001).
In this article, I review existing astrometric observations of X-ray binaries, giving an overview of astrometric VLBI techniques (Section 2), and then focussing on the physical insights that can be derived from both geometric distances (Section 3) and proper motion measurements (Section 4). I examine the use of astrometric residuals to determine orbital parameters or constrain jet sizes (Section 5), and finish with a discussion of the prospects for future progress in this field (Section 6), including not only recent developments in VLBI and the potential contribution of the Square Kilometre Array (SKA), but also the recent launch of the space-based optical astrometric mission, GAIA (Global Astrometric Interferometer for Astrophysics) (Perryman et al. Reference Perryman2001).
2 ASTROMETRIC TECHNIQUES
The theoretical astrometric precision of an interferometer is given by the instrumental resolution divided by twice the signal-to-noise ratio of the detection. For maximum baselines of several thousand kilometres (as for the Very Long Baseline Array (VLBA) or the European VLBI Network) and observing frequencies of a few GHz, then the maximum resolution is on the order of a milliarcsecond. Thus, with a signal-to-noise of 10–20, we can achieve astrometric accuracies of a few tens of microarcseconds. However, in typical astrometric VLBI experiments, these measured positions are not absolute, but measured relative to a nearby (typically extragalactic) background source, in a technique known as phase referencing (see Fomalont Reference Fomalont, Zensus, Diamond and Napier1995, for a detailed overview of astrometric techniques). With sufficient signal-to-noise, the final astrometric precision becomes limited not by statistical uncertainty, but by systematics introduced in interpolating the phase solutions from the nearby phase reference calibrator to the target. Such systematic errors scale linearly with angular separation between science target and calibrator source (Pradel, Charlot, & Lestrade Reference Pradel, Charlot and Lestrade2006), and typically limit the achievable astrometric accuracy to a few tens of microarcseconds.
A typical phase referencing experiment will cycle continuously between a bright, stationary extragalactic background source and the science target of interest. This not only increases the possible integration time beyond the atmospheric coherence time (allowing observations of weak targets), but provides a relative position for the science target relative to that assumed for the calibrator source (Wrobel et al. Reference Wrobel, Walker, Benson and Beasley2000). Successful phase transfer depends on reliably connecting the phases between adjacent scans on the calibrator source (i.e. sufficiently short cycle times) and on the accuracy of the interpolation (i.e. a sufficiently small angular separation between calibrator and target source). Specialised calibration techniques (recently reviewed by Reid & Honma Reference Reid and Honma2013) can be employed to remove uncorrected tropospheric and clock errors from the correlated data using geodetic blocks (occasional short observations of multiple bright calibrators located across the entire sky; Mioduszewski & Kogan Reference Mioduszewski and Kogan2004), or to account for tropospheric phase gradients by observing multiple calibrators close to the target source (Fomalont & Kogan Reference Fomalont and Kogan2005; Fomalont Reference Fomalont, Romney and Reid2005).
Following the transfer of the phases from the nearby calibrator source, the target position may be determined (prior to performing any self-calibration) by fitting a model (typically a point source or a Gaussian) in either the uv- or the image-plane. When imaging, care must be taken with data weighting in the case of an inhomogeneous array. For relatively faint (sub-mJy) sources such as X-ray binaries in the hard or quiescent state, the astrometric accuracy is limited by sensitivity, implying a preference for natural over uniform weighting despite the consequent loss in resolution. However, natural weighting can lead to the measurements being dominated by any systematics affecting the most sensitive baseline(s), so unless required for a detection of the target source, a mild down-weighting of the most sensitive antennas can provide better astrometric accuracy, even at the expense of some signal-to-noise (see the discussions in, e.g. Deller, Tingay, & Brisken Reference Deller, Tingay and Brisken2009; Miller-Jones et al. Reference Miller-Jones, Sivakoff, Knigge, Körding, Templeton and Waagen2013). Except at the peak of the hard state, these sources tend to be unresolved (even with VLBI), and introducing a Gaussian taper into the weighting function is typically unnecessary.
Before fitting for the astrometric parameters, the magnitude of the systematic uncertainties must be assessed and various approaches have been presented in the literature. The simplest is to make occasional observations of an astrometric check source, calibrated in an identical fashion to the science target. Scaling the scatter in its measured positions by its angular separation from the calibrator (relative to that for the target) gives a rough estimate of the magnitude of the systematic errors. Alternatively, for sufficiently bright sources, intra-epoch systematics can be estimated from the positional scatter between different frequency sub-bands (Deller et al. Reference Deller, Tingay and Brisken2009). Alternatively, the full set of measured positions from each frequency sub-band and each epoch can form the sample for a Monte Carlo bootstrapping method of determining the systematics (Chatterjee et al. Reference Chatterjee2009). Finally, the systematics in both right ascension and declination can be adjusted until the final reduced χ2 value of the astrometric fit reaches 1 (e.g. Deller et al. Reference Deller, Tingay and Brisken2009; Reid et al. Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011).
The five basic astrometric parameters are the source reference position and proper motion (in both right ascension and declination; α0, δ0, μαcos δ, μδ), and the source parallax (π). Following Loinard et al. (Reference Loinard, Torres, Mioduszewski, Rodríguez, González-Lópezlira, Lachaume, Vázquez and González2007), the position of a source may be expressed in terms of these five parameters as
\begin{equation}
\begin{array}{rcl} \alpha (t)&=&\alpha _0+(\mu _{\alpha }\cos \delta )t+\pi f_{\alpha }(t),\\[3pt]
\delta (t)&=&\delta _0+\mu _{\delta }t +\pi f_{\delta }(t), \end{array}
\end{equation}
where f α and f δ are the projections of the parallax ellipse onto the right ascension and declination axes (Seidelman Reference Seidelman1992). This set of coupled equations can be solved using a singular value decomposition algorithm (see Loinard et al. Reference Loinard, Torres, Mioduszewski, Rodríguez, González-Lópezlira, Lachaume, Vázquez and González2007, for details). The solution provides the reference position of the source at a given epoch, its proper motion and its parallax, from which the motion of the source on the sky can be determined, as shown in Figure 1.
Figure 1. Astrometric measurements of V404 Cygni over a period of over 4 yr (the time of each epoch is marked on the trace, in years since the first observation). The overall motion is to the southwest, with an annual parallax signature superposed. Deconvolving these two signals allows a measurement of both the parallax and proper motion of the system. Data taken from Miller-Jones et al. (Reference Miller-Jones, Jonker, Dhawan, Brisken, Rupen, Nelemans and Gallo2009a). Adapted from figure 1 of Miller-Jones et al. (Reference Miller-Jones, Jonker, Dhawan, Brisken, Rupen, Nelemans and Gallo2009a) by permission of the AAS.
3 THE X-RAY BINARY DISTANCE SCALE
Distance is a fundamental quantity in astrophysics. Accurate distances are required to convert observational quantities (such as measured fluxes and proper motions) into the corresponding physical quantities (luminosities and speeds, respectively).
Relying on simple geometry alone, trigonometric parallax is the only model-independent method of distance determination, and as such, is the gold standard against which we can calibrate all other methods. However, X-ray binaries are not typically amenable to parallax measurements. Other than a handful of Be/X-ray binaries (Chevalier & Ilovaisky Reference Chevalier and Ilovaisky1998), all known systems are located at ≥1 kpc; the closest known transient neutron star and black hole X-ray binaries are Cen X-4 (≤1.2 ± 0.3 kpc; Chevalier et al. Reference Chevalier, Ilovaisky, van Paradijs, Pedersen and van der Klis1989) and A0620-00 (1.06 ± 0.12 kpc; Cantrell et al. Reference Cantrell2010), respectively. Thus, in the majority of cases, the amplitude of the parallax signal would be less than 1 mas. While this level of accuracy can be achieved with Very Long Baseline Interferometry (VLBI), it requires the source to emit bright, compact radio emission.
The first reported VLBI parallaxes for X-ray binaries were for the two high-mass systems Cygnus X-1 and LSI +61°303 (Lestrade et al. Reference Lestrade, Preston, Jones, Phillips, Rogers, Titus, Rioja and Gabuzda1999), as part of a programme to tie the Hipparcos optical reference frame to the International Celestial Reference Frame (ICRF). While the post-fit residuals were too large to determine the distance to LSI +61°303, the parallax of Cygnus X-1 was detected at the 2.4σ level, giving a distance of 1.4+0.9 −0.4 kpc.
3.1 Reaching the Eddington luminosity
The first truly precise X-ray binary parallax measurement was made for the Z-source neutron star system Sco X-1 (Bradshaw, Fomalont, & Geldzahler Reference Bradshaw, Fomalont and Geldzahler1999). Taking advantage of a nearby (70 arcsec) calibrator to minimise the astrometric systematics, the fitted parallax of 360 ± 40 μas was at the time both the smallest and most precise measurement ever made. The derived distance of 2.8 ± 0.3 kpc proved that Sco X-1 did indeed reach its Eddington luminosity in a particular X-ray spectral state (the vertex of the normal and flaring branches in an X-ray colour-colour diagram). This work was also notable in demonstrating that despite amplitude and structural variations on timescales as short as 10 min, the radio core (assumed to correspond to the binary system itself) could be unambiguously identified in most epochs, thus permitting the required high-precision astrometric measurements.
Distances to neutron star X-ray binaries are often estimated by assuming that certain Type I X-ray bursts (arising from unstable thermonuclear burning of accreted material; see Galloway et al. Reference Galloway, Muno, Hartman, Psaltis and Chakrabarty2008, for a review), known as photospheric radius expansion (PRE) bursts, reach the local Eddington luminosity, so can act as standard candles (to within ~15%; Kuulkers et al. Reference Kuulkers, den Hartog, Zand, Verbunt, Harris and Cocchi2003). However, the expected peak luminosity varies according to the hydrogen mass fraction of the accreted material. The Eddington luminosity for pure He burning is 1.7 times higher than for material of solar composition, corresponding to a 30% change in the estimated distance. Hence, with a sufficiently accurate distance measurement, it could be possible to determine the composition of the accreting material. However, other sources of systematic error could blur this signal, arising from uncertainties in the neutron star mass (which can be at least as high as 2M ⊙; Demorest et al. Reference Demorest, Pennucci, Ransom, Roberts and Hessels2010), the maximum radius reached by the expanding photosphere during the burst (affecting the gravitational redshift and hence the luminosity), and the 5%–10% variation in burst luminosities observed within a given source (Galloway et al. Reference Galloway, Muno, Hartman, Psaltis and Chakrabarty2008).
Although Sco X-1 has never shown Type I X-ray bursts, there are a handful of other neutron star X-ray binaries showing both Type I bursts and detectable radio emission, at least at certain phases of their outburst cycles. Of these, the most nearby sources (with the largest parallax signatures) are the ultracompact system 4U 0614+091 (Kuulkers et al. Reference Kuulkers2010; Migliari et al. Reference Migliari2010), and the atoll sources Aql X-1 (Koyama et al. Reference Koyama1981; Miller-Jones et al. Reference Miller-Jones2010) and 4U 1728-34 (Hoffman et al. Reference Hoffman, Lewin, Doty, Hearn, Clark, Jernigan and Li1976; Migliari et al. Reference Migliari, Fender, Rupen, Jonker, Klein-Wolt, Hjellming and van der Klis2003). Parallax distances to these sources would extend the sample of systems used to calibrate the relationship between PRE burst luminosities and the Eddington luminosity (previously restricted to 12 globular cluster sources; Kuulkers et al. Reference Kuulkers, den Hartog, Zand, Verbunt, Harris and Cocchi2003).
Accurate estimates of the peak outburst luminosity are also important for Galactic black hole X-ray transients, since they can in principle shed light on the nature of ultraluminous X-ray sources (ULXs; see Feng & Soria Reference Feng and Soria2011, for a review), particularly at the low-luminosity end of the ULX luminosity function, around 1039 erg s−1. Although Grimm, Gilfanov, & Sunyaev (Reference Grimm, Gilfanov and Sunyaev2002) found only two transient black hole X-ray binaries in the Galaxy to have exceeded 1039 erg s−1 (the Eddington luminosity for a 10M ⊙ black hole), Jonker & Nelemans (Reference Jonker and Nelemans2004) demonstrated that with more accurate distance estimates, at least five, and possibly up to 7 of the 15 transient Galactic systems could have exceeded this limit, and would thus have been classified as ULXs had they been observed in an external galaxy. Although the majority of ULXs are persistent rather than transient (Feng & Soria Reference Feng and Soria2011), the lack of a break in the ULX luminosity function below 1040 erg s−1 suggests that stellar-mass black holes can indeed exceed the Eddington limit by at least a small factor, a hypothesis borne out by the recent detection of Eddington-rate behaviour in an outburst of a microquasar in our neighbouring galaxy, M31 (Middleton et al. Reference Middleton2013).
To date, only one transient Galactic black hole, V404 Cygni, has an accurate parallax distance measurement (Miller-Jones et al. Reference Miller-Jones, Jonker, Dhawan, Brisken, Rupen, Nelemans and Gallo2009a). This revised the source distance downwards by a factor of 1.7, implying that its 1989 outburst only reached a luminosity of ~0.5 L Edd (see Tanaka & Lewin Reference Tanaka, Lewin, Lewin, van Paradijs and vandenHeuvel1995, for a detailed description of this outburst). While additional parallax measurements for transient sources would be valuable, the short (months-long) durations of their outbursts and the low quiescent luminosities of many systems (Gallo et al. Reference Gallo, Homan, Jonker and Tomsick2008; Miller-Jones et al. Reference Miller-Jones, Jonker, Maccarone, Nelemans and Calvelo2011) preclude radio detections for all but the closest systems (e.g. Gallo et al. Reference Gallo, Fender, Miller-Jones, Merloni, Jonker, Heinz, Maccarone and van der Klis2006). The best candidates for future parallax measurements would therefore be recurrent transients such as H1743-322 or GX339-4 (see Section 6).
3.2 Evidence for event horizons
Accurate luminosities are not only important for X-ray binaries in outburst, but also in quiescence. Black hole X-ray binaries have been found to have systematically lower bolometric luminosities than neutron star systems with similar orbital periods (Narayan, Garcia, & McClintock Reference Narayan, Garcia and McClintock1997; Menou et al. Reference Menou, Esin, Narayan, Garcia, Lasota and McClintock1999; Garcia et al. Reference Garcia, McClintock, Narayan, Callanan, Barret and Murray2001), which was attributed to the existence of radiatively inefficient accretion flows (RIAFs) and event horizons in the black hole systems. Although this interpretation has been challenged (Campana & Stella Reference Campana and Stella2000; Abramowicz, Kluźniak, & Lasota Reference Abramowicz, Kluźniak and Lasota2002), and alternative explanations are possible (e.g. coronal emission; Bildsten & Rutledge Reference Bildsten and Rutledge2000, energy being channeled into jets; Fender, Gallo, & Jonker Reference Fender, Gallo and Jonker2003), the original claim also relies on accurate estimates of the source luminosities, and hence distances.
3.3 Determining system parameters
Accurate distances to black hole X-ray binaries can also be invaluable in constraining the physical parameters of the binary system. In quiescent systems, an accurate distance can be used to determine the luminosity of the hotspot where the accretion stream impacts the disc, allowing the mass transfer rate from the secondary star to be determined (e.g. Froning et al. Reference Froning2011), providing important observational constraints for binary evolution models.
Other key system parameters that can benefit from accurate distance determinations are the component masses and orbital inclination angle. Using a larger bandwidth and an observational strategy designed to minimise systematic uncertainties (such as the use of geodetic blocks), Reid et al. (Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011) revisited the parallax of Cygnus X-1, finding a distance of 1.86+0.12 −0.11 kpc; consistent with but significantly more precise than the previous measurement of Lestrade et al. (Reference Lestrade, Preston, Jones, Phillips, Rogers, Titus, Rioja and Gabuzda1999). The source distance had been a major uncertainty in determining the system parameters of this persistent X-ray binary (e.g. Paczynski Reference Paczynski1974), which contains the first black hole to be discovered, and has since become one of the most well-studied black hole systems, providing important insights into accretion physics.
With the new parallax distance, accurate to 6%, Orosz et al. (Reference Orosz, McClintock, Aufdenberg, Remillard, Reid, Narayan and Gou2011) were able to determine the donor star radius from its K-band magnitude, thereby strongly constraining the dynamical model for the system. Adding in other constraints (radial velocity curves and optical photometry) they determined the black hole and donor masses, the inclination angle of the orbital plane, and measured a non-zero eccentricity for the orbit.
The uncertain distance for GRS 1915+105 also provides the bulk of the uncertainty in determining its system parameters (McClintock et al. Reference McClintock, Shafee, Narayan, Remillard, Davis and Li2006), which have recently been revised by Steeghs et al. (Reference Steeghs, McClintock, Parsons, Reid, Littlefair and Dhillon2013). The latter authors are already undertaking an astrometric programme to determine a parallax distance to the source, the results of which should finally pin down the nature of this enigmatic system, which has almost certainly been accreting close to the Eddington rate for over two decades, and has provided an ideal laboratory for studying disc-jet coupling (see Fender & Belloni Reference Fender and Belloni2004, for a review).
3.4 Constraining black hole spin
With accurate values of distance, inclination angle and black hole mass, it is possible to fit high-quality, disc-dominated X-ray spectra of black hole X-ray binaries with fully relativistic models for the accretion disc to measure the black hole spin (see McClintock, Narayan, & Steiner Reference McClintock, Narayan and Steiner2013, for a review). Since the black hole spin sets the radius of the innermost stable circular orbit (ISCO), then assuming that the accretion disc is sharply truncated at the ISCO, the spin can be determined from the fitted inner disc radius. This method has so far been used to determine the spins of 10 stellar-mass black holes. However, it relies on accurate pre-existing constraints on the source distance, inclination angle, and black hole mass, and uncertainties in these parameters are the major source of uncertainty in the derived spins.
Using the accurate values of distance (Reid et al. Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011), inclination, and mass (Orosz et al. Reference Orosz, McClintock, Aufdenberg, Remillard, Reid, Narayan and Gou2011) measured for Cygnus X-1, Gou et al. (Reference Gou2011) were able to measure an extremely high value for the dimensionless spin parameter of a *>0.95, in good agreement with a recent measurement derived from an analysis of the relativistically-broadened Fe Kα line profile (Duro et al. Reference Duro2011). Such a high spin is believed to be shared only by GRS 1915+105 (McClintock et al. Reference McClintock, Shafee, Narayan, Remillard, Davis and Li2006; Blum et al. Reference Blum, Miller, Fabian, Miller, Homan, van der Klis, Cackett and Reis2009) among the black hole X-ray binaries, and, if it can be tapped by the Blandford-Znajek mechanism (Blandford & Znajek Reference Blandford and Znajek1977), implies the possibility of extremely powerful jets.
It has recently been claimed that ballistic jets from transient black hole X-ray binaries that reach a significant fraction of their Eddington limit are indeed powered by black hole spin (Narayan & McClintock Reference Narayan and McClintock2012; Steiner, McClintock, & Narayan Reference Steiner, McClintock and Narayan2013). This claim relies on an apparent correlation between the measured spins of selected transient black hole X-ray binaries and a proxy for their jet powers (the maximum unbeamed 5 GHz radio luminosity during outburst, scaled by the black hole mass). However, this remains controversial (Fender, Gallo, & Russell Reference Fender, Gallo and Russell2010; Russell, Gallo, & Fender Reference Russell, Gallo and Fender2013), owing to the difficulty in identifying an accurate proxy for the jet power, the inherent uncertainties on the measured black hole spins, and the small number of sources deemed to be suitable for inclusion in the sample. More accurate distance measurements from VLBI parallaxes would help to reduce the uncertainties in the measured spins and jet powers, thereby helping to resolve this important debate.
3.5 The neutron star equation of state
As discussed by Tomsick et al. (Reference Tomsick, Quirrenbach, Kulkarni, Shaklan and Pan2009), accurate distances to neutron star systems can also help constrain the neutron star equation of state (Lattimer & Prakash Reference Lattimer and Prakash2007). Different equations of state produce different mass–radius relationships, and although recent years have seen some extremely accurate neutron star mass measurements (e.g. Demorest et al. Reference Demorest, Pennucci, Ransom, Roberts and Hessels2010), radius determinations are often dependent on the source distance. The quiescent X-ray emission from neutron star X-ray binaries is dominated by the blackbody emission from the neutron star surface. A model-independent geometric distance measurement would allow the luminosity to be determined more accurately (to the accuracy of the X-ray flux scale, typically 10%–20%), allowing the area (and hence the radius) of the emitter to be determined via the Stefan–Boltzmann law. A precise determination of both mass and radius for just a single neutron star would be invaluable in ruling out many of the proposed equations of state.
4 COMPACT OBJECT FORMATION AND NATAL KICKS
Even for objects whose distances are too great, or for which systematic astrometric uncertainties are too large to measure a parallax distance, it is possible to measure a proper motion, since the signal is cumulative with time. If the source distance and the systemic radial velocity can also be determined (the latter typically from optical or near-infrared spectroscopy), then all six position and velocity components are known. By integrating backwards in time in the Galactic potential, it is possible to trace the orbit of the system through the Galaxy (e.g. Figure 2), and hence derive constraints on the formation of the compact object.
Figure 2. Tracing back the trajectory of V404 Cygni through the potential of the Galaxy over the past 1 Gyr (adapted from figure 3 of Miller-Jones et al. Reference Miller-Jones, Jonker, Nelemans, Portegies Zwart, Dhawan, Brisken, Gallo and Rupen2009b, using the updated astrometric parameters of Miller-Jones et al. Reference Miller-Jones, Jonker, Dhawan, Brisken, Rupen, Nelemans and Gallo2009a). The white circle marks the current position of the Sun, and the white triangle the current position of V404 Cygni. The red trace shows the past trajectory for the best-fitting astrometric parameters, with the white (a,b,c) or grey (d,e) traces showing the range of possible trajectories within the 1σ uncertainties. In panels a–c, the greyscale shows the mass density (assuming the Galactic potential of Johnston et al. Reference Johnston, Spergel and Hernquist1995). Panels (c) and (d) show zoomed-in versions of (b) and (a), respectively. In its orbit around the Galactic centre (marked with a cross), the vertical trajectory of V404 Cygni never reaches more than ~100 pc above the Galactic Plane.
4.1 Natal kicks
The high space velocities of radio pulsars provide good evidence for strong natal kicks during the formation of neutron stars (Lyne & Lorimer Reference Lyne and Lorimer1994). These kicks, which can give rise to velocities in excess of 1000 km s−1 (Hobbs et al. Reference Hobbs, Lorimer, Lyne and Kramer2005), cannot be explained purely by the supernova recoil kick (Blaauw Reference Blaauw1961). The recoil is set by the ejected mass, and since ejection of more than half the total mass causes a binary system to become unbound, this sets an upper limit to the maximum recoil velocity (Nelemans, Tauris, & van den Heuvel Reference Nelemans, Tauris and van den Heuvel1999). Alternative possibilities for generating high natal kick velocities typically involve hydrodynamical mechanisms, asymmetric neutrino emission induced by strong magnetic fields, or electromagnetic kicks from an off-centre rotating dipole, and have been reviewed in detail by Lai (Reference Lai, Blaschke, Glendenning and Sedrakian2001).
A second population of neutron stars is believed to form with significantly lower kicks (Pfahl, Rappaport, & Podsiadlowski Reference Pfahl, Rappaport and Podsiadlowski2002a; Pfahl et al. Reference Pfahl, Rappaport, Podsiadlowski and Spruit2002b), potentially due to a smaller iron core in the progenitor star, or to formation in an electron-capture supernova (Podsiadlowski et al. Reference Podsiadlowski, Langer, Poelarends, Rappaport, Heger and Pfahl2004). The ensuing prompt or fast explosion does not allow time for convectively-driven instabilities to grow in the neutrino-heated layer behind the supernova shock (Scheck et al. Reference Scheck, Plewa, Janka, Kifonidis and Müller2004), leading to smaller kick velocities. The recent discovery of two distinct subpopulations of Be/X-ray binaries provides further observational support for a dichotomy between these two types of supernova (Knigge, Coe, & Podsiadlowski Reference Knigge, Coe and Podsiadlowski2011).
Black holes are believed to form in two different ways (Fryer & Kalogera Reference Fryer and Kalogera2001). For a sufficiently massive progenitor, they may form by direct collapse. Alternatively, if a supernova explosion is not sufficiently energetic to unbind the stellar envelope, fallback of ejected material onto the proto-neutron star formed in the explosion can create a black hole. In the latter case, many of the non-recoil kick mechanisms that have been proposed for neutron stars (with the exception of the electromagnetic kicks) could also apply to black holes.
The similarity in the distributions of black hole and neutron star X-ray binary systems with Galactic latitude has been used to argue for equivalent natal kicks during black hole formation (Jonker & Nelemans Reference Jonker and Nelemans2004). Indeed, detailed population synthesis calculations have suggested (albeit discounting observational selection effects) that such kicks are necessary, with the magnitudes of black hole kick velocities (rather than their momenta) being similar to those of neutron stars (Repetto, Davies, & Sigurdsson Reference Repetto, Davies and Sigurdsson2012). This latter point could be used to discriminate between proposed kick mechanisms; while neutrino-driven kicks should give rise to the same momenta in black holes and neutron stars, hydrodynamical kicks from asymmetries in the supernova ejecta can accelerate a nascent black hole to similarly high velocities as observed in neutron stars (Janka Reference Janka2013).
Thus, VLBI measurements of the proper motions of black hole X-ray binaries can be used to probe the black hole formation mechanism, determining whether or not a natal kick is required for a given system, and, eventually, determining the distribution of black hole kick velocities. A bimodal distribution would be good evidence for some black holes to form without a natal supernova, with the most massive black holes (not having lost material in the explosion) likely to have the lowest velocities relative to their local standard of rest (LSR).
4.2 Observational constraints
4.2.1 Black holes
The first evidence for a strong natal kick in a black hole system was found from optical spectroscopy of GRO J1655-40. Brandt, Podsiadlowski, & Sigurdsson (Reference Brandt, Podsiadlowski and Sigurdsson1995) considered possible explanations for the large measured systemic radial velocity of 150 ± 19 km s−1 (Bailyn et al. Reference Bailyn, Orosz, McClintock and Remillard1995), including rocket acceleration by jets, a triple system, discrete scattering events, or perturbations due to interactions with density waves in the Galactic potential. These were all deemed to be unlikely, leaving a natal kick in a supernova explosion as the most plausible explanation (a scenario that is also supported by the observed misalignment between the disc plane and the jet axis; Maccarone Reference Maccarone2002). With the subsequent measurement of a proper motion for the system by the Hubble Space Telescope (Mirabel et al. Reference Mirabel, Mignani, Rodrigues, Combi, Rodríguez and Guglielmetti2002), more detailed modelling was able to reconstruct the full evolutionary history of the binary system since the black hole was formed (Willems et al. Reference Willems, Henninger, Levin, Ivanova, Kalogera, McGhee, Timmes and Fryer2005). Although formation with no natal kick could not be formally excluded, an asymmetric supernova explosion was found to be most likely, imparting a kick of 45–115 km s−1 to the binary, and giving rise to an eccentric orbit in the plane of the Galaxy.
In the case of XTE J1118+480, an even more compelling case for a natal kick could be made from the measured proper motion (Mirabel et al. Reference Mirabel, Dhawan, Mignani, Rodrigues and Guglielmetti2001). The derived space velocity of 145 km s−1 relative to the LSR implied that the system was on a halo orbit, consistent with either an extraordinarily large natal kick, or formation in a globular cluster (although the latter explanation was subsequently ruled out by the supersolar chemical abundances of the donor star; González Hernández et al. Reference González Hernández, Rebolo, Israelian, Harlaftis, Filippenko and Chornock2006). By supplementing this information with the known system parameters (component masses, orbital period, donor star properties), Fragos et al. (Reference Fragos, Willems, Kalogera, Ivanova, Rockefeller, Fryer and Young2009) were able to demonstrate that this system must have been formed with a natal kick of between 80 and 310 km s−1 (see also Gualandris et al. Reference Gualandris, Colpi, Zwart and Possenti2005).
In contrast to these low-mass X-ray binaries, the high-mass system Cygnus X-1 was measured to have a relatively low proper motion (Chevalier & Ilovaisky Reference Chevalier and Ilovaisky1998; Mirabel & Rodrigues Reference Mirabel and Rodrigues2003a; Reid et al. Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011). While the high mass of the companion should reduce the recoil velocity of the system, the observed proper motion can be explained without an asymmetric natal kick, either by symmetric mass ejection in a supernova (Nelemans et al. Reference Nelemans, Tauris and van den Heuvel1999), or via direct collapse into a black hole (Mirabel & Rodrigues Reference Mirabel and Rodrigues2003a). More detailed modelling was performed by Wong et al. (Reference Wong, Valsecchi, Fragos and Kalogera2012), who used the more recent observational constraints of Reid et al. (Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011) and Orosz et al. (Reference Orosz, McClintock, Aufdenberg, Remillard, Reid, Narayan and Gou2011), and were able to place an upper limit of 77 km s−1 on the natal kick velocity.
The only other measured black hole proper motions also suggest relatively small natal kicks. Dhawan et al. (Reference Dhawan, Mirabel, Ribó and Rodrigues2007) determined the proper motion of GRS 1915+105, which, combined with the best available systemic radial velocity of −3 ± 10 km s−1 (Greiner, Cuby, & McCaughrean Reference Greiner, Cuby and McCaughrean2001), they used to determine its peculiar velocity as a function of the unknown source distance. The proper motions of the jet ejecta during outbursts imply a maximum source distance of 11–12 kpc (Mirabel & Rodríguez Reference Mirabel and Rodríguez1994; Fender et al. Reference Fender, Garrington, McKay, Muxlow, Pooley, Spencer, Stirling and Waltman1999), although a possible association with two IRAS sources has been used to argue for a distance of order 6 kpc (Kaiser et al. Reference Kaiser, Gunn, Brocksopp and Sokoloski2004). Dhawan et al. (Reference Dhawan, Mirabel, Ribó and Rodrigues2007) found the peculiar velocity to be minimised for a distance of 9–10 kpc, and to be <83 km s−1 even for the maximum possible distance of 12 kpc, and therefore concluded that no natal supernova kick was required. Similarly, V404 Cygni (M = 9+0.2 −0.6 M ⊙; Khargharia, Froning, & Robinson Reference Khargharia, Froning and Robinson2010), was found to have a peculiar velocity of 65 km s−1 (Miller-Jones et al. Reference Miller-Jones, Jonker, Dhawan, Brisken, Rupen, Nelemans and Gallo2009a, Reference Miller-Jones, Jonker, Nelemans, Portegies Zwart, Dhawan, Brisken, Gallo and Rupen2009b), which could be explained purely by a recoil kick from a natal supernova.
The measured astrometric parameters of these five black hole systems are presented in Table 1. However, a comparison of their peculiar velocities and the significance of any correlation with black hole mass is complicated by the differing assumptions made regarding the distance of the Sun from the Galactic centre (R 0), the rotational velocity of the Galaxy (Θ0), and the solar motion with respect to the LSR, (U ⊙,V ⊙,W ⊙). Using the values of R 0 = 8.05 ± 0.45 kpc and Θ0 = 238 ± 14 km s−1 determined by Honma et al. (Reference Honma2012), and the solar motion of (U ⊙,V ⊙,W ⊙) = (11.1+0.69 −0.75, 12.24+0.47 −0.47, 7.25+0.37 −0.36) km s−1 measured by Schönrich, Binney, & Dehnen (Reference Schönrich, Binney and Dehnen2010), we have therefore applied the transformations of Johnson & Soderblom (Reference Johnson and Soderblom1987) to determine the full three-dimensional space velocity of each system, and used this to derive their peculiar velocities (Table 2).
Table 1 Measured astrometric parameters of X-ray binaries. Systems have been divided into confirmed black holes (top) and neutron stars (middle), and systems whose compact object is still unknown (bottom).
a This distance is derived from the proper motions of the relativistic jets, assuming an inclination angle for the system of 85 ± 2°. Other authors have suggested a closer distance (Mirabel et al. Reference Mirabel, Mignani, Rodrigues, Combi, Rodríguez and Guglielmetti2002; Foellmi Reference Foellmi2009).
b Although a distance of 11 ± 1 kpc is favoured should the systemic velocity track Galactic rotation, and also from the proper motions of relativistic jets, Kaiser et al. (Reference Kaiser, Gunn, Brocksopp and Sokoloski2004) have suggested distances as low as 6 kpc.
c The quoted distance is for the accretion of material of solar metallicity onto the donor star; accretion of helium-rich material would give a distance higher by a factor of 1.3 (Galloway et al. Reference Galloway, Muno, Hartman, Psaltis and Chakrabarty2008).
d Proper motions have been deduced from the VLBI positions reported by Miller-Jones et al. (Reference Miller-Jones2010) and Tudose et al. (Reference Tudose, Paragi, Yang, Miller-Jones, Fender, Garrett, Rushton and Spencer2013).
References: [1] Mirabel et al. (Reference Mirabel, Dhawan, Mignani, Rodrigues and Guglielmetti2001); [2] Gelino et al. (Reference Gelino, Balman, Kızıloǧlu, Yılmaz, Kalemci and Tomsick2006); [3] González Hernández et al. (Reference González Hernández, Rebolo, Israelian, Filippenko, Chornock, Tominaga, Umeda and Nomoto2008); [4] Mirabel et al. (Reference Mirabel, Mignani, Rodrigues, Combi, Rodríguez and Guglielmetti2002); [5] Hjellming & Rupen (Reference Hjellming and Rupen1995); [6] Shahbaz et al. (Reference Shahbaz, van der Hooft, Casares, Charles and van Paradijs1999); [7] Dhawan et al. (Reference Dhawan, Mirabel, Ribó and Rodrigues2007); [8] Steeghs et al. (Reference Steeghs, McClintock, Parsons, Reid, Littlefair and Dhillon2013); [9] Reid et al. (Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011); [10] Gies et al. (Reference Gies2008); [11] Miller-Jones et al. (Reference Miller-Jones, Jonker, Dhawan, Brisken, Rupen, Nelemans and Gallo2009a); [12] Casares & Charles (Reference Casares and Charles1994); [13]Dhawan, Mioduszewski, & Rupen (Reference Dhawan, Mioduszewski, Rupen and Belloni2006); [14] Frail & Hjellming (Reference Frail and Hjellming1991); [15] Casares et al. (Reference Casares, Ribas, Paredes, Martí and Allende Prieto2005); [16] González Hernández et al. (Reference González Hernández, Rebolo, Peñarrubia, Casares and Israelian2005); [17] Casares et al. (Reference Casares, Bonifacio, González Hernández, Molaro and Zoccali2007); [18] Bradshaw et al. (Reference Bradshaw, Fomalont and Geldzahler1999); [19] Steeghs & Casares (Reference Steeghs and Casares2002); [20] Moldon et al. (Reference Moldon2012); [21] Casares et al. (Reference Casares, Ribas, Paredes, Martí and Allende Prieto2005b); [22] Miller-Jones et al. (Reference Miller-Jones2010); [23] Tudose et al. (Reference Tudose, Paragi, Yang, Miller-Jones, Fender, Garrett, Rushton and Spencer2013); [24] Galloway et al. (Reference Galloway, Muno, Hartman, Psaltis and Chakrabarty2008); [25] Cornelisse et al. (Reference Cornelisse, Casares, Steeghs, Barnes, Charles, Hynes and O'Brien2007); [26] Spencer et al. (Reference Spencer, Rushton, Bałucińska-Church, Paragi, Schulz, Wilms, Pooley and Church2013); [27] Elebert et al. (Reference Elebert, Callanan, Torres and Garcia2009); [28]Lockman, Blundell, & Goss (Reference Lockman, Blundell and Goss2007); [29] Blundell & Bowler (Reference Blundell and Bowler2004); [30] Hillwig et al. (Reference Hillwig, Gies, Huang, McSwain, Stark, van der Meer and Kaper2004); [31] Miller-Jones et al. (Reference Miller-Jones, Sakari, Dhawan, Tudose, Fender, Paragi and Garrett2009c); [32] Ling, Zhang, & Tang (Reference Ling, Zhang and Tang2009).
Table 2 Inferred Galactic space velocities of X-ray binaries.
U, V, and W are defined as positive towards l = 0°, l = 90°, and b = 90°, respectively. UC and VC are the velocities expected from circular rotation at 238 km s−1 (Honma et al. Reference Honma2012). The peculiar velocity v pec is defined as [(U−UC)2 + (V−VC)2 + W 2]1/2.
With such a small sample, it is not possible to conclusively determine whether the kick velocity correlates with compact object (or total system) mass (i.e. to discriminate between momentum-conserving and velocity-conserving kicks), or whether there is a clear mass dichotomy between black holes forming by direct collapse and those forming in a natal supernova. The masses of the black holes with accurate astrometric data are given in Table 3, and plotted against derived peculiar velocity in Figure 3. While the three lowest peculiar velocities are associated with the three highest-mass black holes, we note that more recent Bayesian methods of constraining the black hole masses (Farr et al. Reference Farr, Sravan, Cantrell, Kreidberg, Bailyn, Mandel and Kalogera2011; Kreidberg et al. Reference Kreidberg, Bailyn, Farr and Kalogera2012) suggest that XTE J1118+480 and V404 Cyg have almost identical black hole masses. The difference in their peculiar velocities suggests that kick velocity and black hole mass may not be directly related.
Table 3 Measured black hole masses and peculiar velocities.
Peculiar velocities taken from Table 2. References for the black hole mass: [1] Khargharia et al. (Reference Khargharia, Froning, Robinson and Gelino2013); [2] Beer & Podsiadlowski (Reference Beer and Podsiadlowski2002); [3] Steeghs et al. (Reference Steeghs, McClintock, Parsons, Reid, Littlefair and Dhillon2013); [4] Orosz et al. (Reference Orosz, McClintock, Aufdenberg, Remillard, Reid, Narayan and Gou2011); [5] Khargharia et al. (Reference Khargharia, Froning and Robinson2010).
Figure 3. Inferred peculiar velocity as a function of black hole mass. Black points denote low-mass X-ray binaries, and the red point represents the high-mass X-ray binary Cygnus X-1. A larger sample is required to make robust inferences about any potential correlation between black hole (or companion) mass and natal kicks.
4.2.2 Neutron stars
Although neutron stars in X-ray binaries should have formed in a supernova explosion, their existence in a binary system implies that their natal kicks should typically have been lower than those deduced for the radio pulsar population, since a sufficiently strong natal kick would unbind the binary. Indeed, the population of recycled pulsars (spun up due to accretion from a binary companion) is observed to have a significantly lower mean space velocity than that of normal pulsars (Hobbs et al. Reference Hobbs, Lorimer, Lyne and Kramer2005).
Some of the first X-ray binary proper motions were determined from Hipparcos data for a sample of high-mass neutron star systems (Chevalier & Ilovaisky Reference Chevalier and Ilovaisky1998). They showed that the mean transverse velocities of Be/X-ray binaries were lower than those of supergiant systems. Optical observations have also been used to determine the proper motion of Cen X-4 (González Hernández et al. Reference González Hernández, Rebolo, Peñarrubia, Casares and Israelian2005, see Table 1).
Owing to their intrinsically fainter radio emission (Migliari & Fender Reference Migliari and Fender2006), only a handful of neutron star X-ray binaries are accessible to the higher precision astrometry possible using VLBI. Of the five systems with VLBI proper motions, two (Sco X-1 and Cyg X-2) are Z-sources, persistently accreting at or close to the Eddington rate, and showing resolved radio jets (Fomalont, Geldzahler, & Bradshaw Reference Fomalont, Geldzahler and Bradshaw2001; Spencer et al. Reference Spencer, Rushton, Bałucińska-Church, Paragi, Schulz, Wilms, Pooley and Church2013). The atoll source, Aql X-1, was observed during two of its transient outbursts (Miller-Jones et al. Reference Miller-Jones2010; Tudose et al. Reference Tudose, Paragi, Yang, Miller-Jones, Fender, Garrett, Rushton and Spencer2013), when it showed only marginal evidence for resolved jets. The remaining two systems are gamma-ray binaries (LS 5039 and LSI +61°303), whose radio emission is instead likely to arise from a collision between the relativistic wind of a pulsar and the stellar wind of its companion star.
Of these systems, Sco X-1, Cen X-4, and LS 5039 were all deemed to have undergone a strong natal kick at formation (Mirabel & Rodrigues Reference Mirabel and Rodrigues2003b; González Hernández et al. Reference González Hernández, Rebolo, Peñarrubia, Casares and Israelian2005; Moldon et al. Reference Moldon2012), and despite its low peculiar velocity, the high eccentricity of LSI +61°303 is a strong indication that it should also have received an asymmetric kick in the natal supernova (Dhawan et al. Reference Dhawan, Mioduszewski, Rupen and Belloni2006). This leaves Aql X-1 as the only neutron star system in our sample (Table 2) that is unlikely to have received a strong natal kick.
4.2.3 Neutron stars and black holes: a comparison
Although the Galactic distribution of neutron star and black hole X-ray binaries suggests similar natal kicks for neutron star and black hole systems (Jonker & Nelemans Reference Jonker and Nelemans2004), we can directly test this by determining whether the neutron star and black hole peculiar velocities in Table 2 are drawn from the same underlying distribution. Despite the three systems with the highest peculiar velocities being neutron stars, a Kolmogorov–Smirnov test suggests that the null hypothesis cannot be ruled out at better than the 63% level. Thus there is no statistically significant difference between the current samples. However, those samples are small and were selected primarily (but not exclusively) on the basis of radio brightness. Thus, a meaningful test of this hypothesis requires further astrometric measurements.
4.3 Birthplaces
For the youngest systems (i.e. the high-mass X-ray binaries), detailed position and velocity information can allow us to determine the birthplace of the compact object. Based on the low peculiar velocity of Cygnus X-1 and the similarities between its proper motion and that of the nearby star cluster Cygnus OB3, Mirabel & Rodrigues (Reference Mirabel and Rodrigues2003a) suggested that Cygnus X-1 had originated in this star cluster. However, in the case of Cygnus X-3, an unknown compact object in orbit with a WN7 Wolf-Rayet star (van Kerkwijk et al. Reference van Kerkwijk1992), the high mass loss rate in the stellar wind has to date precluded the identification of optical lines from the disc or companion star, such that the systemic radial velocity is poorly constrained (|γ|<200 km s−1). Nevertheless, Miller-Jones et al. (Reference Miller-Jones, Sakari, Dhawan, Tudose, Fender, Paragi and Garrett2009c) used archival Very Large Array (VLA) and VLBA data to determine the proper motion of the system, and inferred a peculiar velocity in the range 9–250 km s−1. Although the Wolf–Rayet companion implies that the system must be relatively young, no potential progenitor star cluster could be identified owing to the uncertain systemic radial velocity and the high extinction along the line of sight.
Astrometric measurements have also shed light on the origin of the persistent, super-Eddington system SS 433. It is offset a few parsecs to the west of the centre of the W50 nebula (Lockman et al. Reference Lockman, Blundell and Goss2007) that is believed to be the supernova remnant (SNR) arising from the creation of the compact object. Its measured three-dimensional space velocity is of order 35 km s−1, oriented back towards the Galactic plane, suggesting that the original binary system was originally ejected from the Galactic plane. The compact object progenitor then underwent a supernova explosion within the past 105 yr, giving rise to a small natal kick that can account for the current peculiar velocity and offset from the centre of W50 (Lockman et al. Reference Lockman, Blundell and Goss2007).
The proximity of the neutron star system Circinus X-1 to the supernova remnant G321.9-0.3 led to similar suggestions, that this SNR was the birthplace of the X-ray binary (Clark, Parkinson, & Caswell Reference Clark, Parkinson and Caswell1975). However, the HST upper limit on the proper motion of the system subsequently ruled out this scenario (Mignani et al. Reference Mignani, De Luca, Caraveo and Mirabel2002). This conclusion was recently confirmed by the detection of a faint X-ray nebula surrounding the X-ray binary, identified (together with the associated radio nebula) as the supernova remnant from the formation of the neutron star, placing an upper limit on its age of 4600 yr (Heinz et al. Reference Heinz2013).
Assuming that black holes formed in the Galactic plane (as inferred in several cases from the chemical abundances of their secondary stars; e.g. González Hernández et al. Reference González Hernández, Rebolo, Israelian, Filippenko, Chornock, Tominaga, Umeda and Nomoto2008), accurate three-dimensional space velocities for black holes can also provide lower limits on their ages. By tracing their trajectories back in time in the Galactic potential (e.g. Figure 2), the times at which they crossed the plane can be determined. The most recent crossing that also satisfies constraints from binary evolution modelling then provides a lower limit on the age of the black hole (e.g. Fragos et al. Reference Fragos, Willems, Kalogera, Ivanova, Rockefeller, Fryer and Young2009).
Finally, for those X-ray binaries detected within globular clusters, the natal kicks must have been sufficiently small for the systems to remain bound to the host cluster (see, e.g. Pfahl et al. Reference Pfahl, Rappaport and Podsiadlowski2002a). Astrometric proper motion measurements could both confirm an association with the cluster, and allow us to probe the movements of the systems within the cluster potential, improving our understanding of the intracluster dynamics.
5 ASTROMETRIC RESIDUALS
Having fit a set of positional measurements to determine the proper motion and parallax of the target source, the astrometric residuals contain additional information on the size scales of both the jets and the binary orbit, which can be probed using sufficiently precise measurements.
5.1 Orbital phase-resolved astrometry
The radio emission detected by VLBI typically arises from a steady, compact, partially self-absorbed radio jet (Blandford & Königl Reference Blandford and Königl1979), likely launched from a few tens of gravitational radii from the black hole. With sufficiently high-precision astrometry, the orbital signature of the black hole around its companion can be determined. The size of this orbital signature is given by
\begin{equation}
r_{\rm BH} = \frac{M_{\rm d}}{(M_{\rm BH}+M_{\rm d})^{2/3}}\left(\frac{GP^2}{4\pi ^2}\right)^{1/3},
\end{equation}
where P is the orbital period, and M BH and M d are the masses of the black hole and donor star, respectively. This implies that such measurements are likely to be successful only for systems with high-mass companions and long orbital periods. Should this be feasible, however, it can provide independent constraints on the system parameters. In the only example to date, Reid et al. (Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011) were able to measure the orbital signature of the black hole in Cygnus X-1, with the reduced χ2 values favouring a clockwise orbit (see also Figure 4(b)). In this case, the system parameters of Orosz et al. (Reference Orosz, McClintock, Aufdenberg, Remillard, Reid, Narayan and Gou2011) were supplied to determine the magnitude of the orbital signature, but with sufficiently high-precision measurements, astrometric data could be used to constrain this independently.
Figure 4. Astrometric residuals in Cygnus X-1.
In neutron star systems, where the mass ratio q 1 = M x/M d is smaller, the orbital signature should be easier to determine. In a sequence of 12 VLBA observations, sampling the full 26-d orbit of the gamma-ray binary LSI +61°303, Dhawan et al. (Reference Dhawan, Mioduszewski, Rupen and Belloni2006) showed that the measured source position traced out an elliptical locus on the plane of the sky, interpreted as the orbital signature of the source. However, the measured size of the ellipse was found to be significantly larger at 2.3 GHz than at 8.4 GHz, and in both cases was much larger than the size of the orbit inferred from the measured system parameters. Furthermore, the position angle of the 2.3-GHz emission trailed that of the higher-frequency emission, leading Dhawan et al. (Reference Dhawan, Mioduszewski, Rupen and Belloni2006) to suggest that they were observing emission from an extended cometary tail that trailed the orbit of the neutron star, hence favouring a pulsar wind origin for the observed radio emission (as in PSR B1259-63) over a precessing radio jet. Although a reanalysis of the same data by Massi et al. (Reference Massi, Ros and Zimmermann2012) was used to argue for the precessing jet scenario, follow-up observations by Moldon (Reference Moldon2012) have provided further evidence for the pulsar wind model.
5.2 Core shifts and the size scale of the jets
Having subtracted off the parallax, proper motion, and orbital motion signatures from the measured source positions, the astrometric residuals should lie along the axis of the radio jets (Figure 4(c)). In a classical partially self-absorbed jet, we see emission from the surface where the optical depth is unity at the observing frequency. This implies that lower-frequency emission arises from further downstream, giving rise to a core shift, as frequently observed in active galactic nuclei (AGN, e.g. Lobanov Reference Lobanov1998; Kovalev et al. Reference Kovalev, Lobanov, Pushkarev and Zensus2008). Measurement of the core shift as a function of frequency is a well-established technique in AGN, where the lack of a moving source makes such measurements easier, and can be used to determine the structure of the jet.
Since astrometric positions are measured relative to the assumed position of a nearby background calibrator, which may differ at different frequencies (owing to its own core shift), core shift measurements require careful astrometry. For the gamma-ray binary LSI +61°303, Moldon (Reference Moldon2012) has shown that when using multiple extragalactic calibrator sources it is possible to disentangle the core shifts of both the calibrators and the target source, although such a technique has not to date been applied to any other X-ray binaries.
5.3 Jet orientation and extent
The compact jets seen in the hard and quiescent states, which form the astrometric targets for VLBI observations, are known to be variable (e.g. Miller-Jones et al. Reference Miller-Jones, Jonker, Dhawan, Brisken, Rupen, Nelemans and Gallo2009a). Small-scale flaring events can arise from changes in the velocity, magnetic field strength, or electron density at the base of the jets. This can affect the position of the τ = 1 surface, causing it to move up or downstream along the jet axis. The astrometric residuals from such compact jets at a range of different brightnesses can therefore be used to determine the orientation of the jet axis and the extent of the jets as a function of frequency, once the proper motion, parallax, and orbital signatures have been removed. Rushton et al. (Reference Rushton2012) used the previously-determined astrometric parameters of Cygnus X-1 (Reid et al. Reference Reid, McClintock, Narayan, Gou, Remillard and Orosz2011) to infer the existence of a remnant compact jet from the astrometric residuals as the source began a transition to its softer X-ray spectral state (when the compact jets are usually believed to be quenched). The astrometric residuals were seen to align with the well-known jet axis from VLBI imaging (Stirling et al. Reference Stirling, Spencer, de la Force, Garrett, Fender and Ogley2001, see also Figure 4(a)), and showed more scatter at 2.3 GHz than at 8.4 GHz, easily explained in the scenario where the jets are more extended at lower frequencies owing to the τ = 1 surface being further downstream (Figure 4(c)).
6 FUTURE PROSPECTS
The majority of persistent, bright X-ray binaries have already been the targets of astrometric observations (Table 1). Since astrometric observations with VLBI are only possible in the hard spectral state seen at the beginning and end of an outburst, and for the closest or brightest quiescent systems (Gallo et al. Reference Gallo, Fender, Miller-Jones, Merloni, Jonker, Heinz, Maccarone and van der Klis2006; Miller-Jones et al. Reference Miller-Jones, Jonker, Maccarone, Nelemans and Calvelo2011), relatively few easily-accessible targets remain. While proper motions can be measured from hard state observations at the start and end of a single X-ray binary outburst (e.g. Mirabel et al. Reference Mirabel, Dhawan, Mignani, Rodrigues and Guglielmetti2001), accurate parallax measurements require observations across the full parallax ellipse, particularly at the times of maximum and minimum parallax displacement. Since the timing of X-ray binary outbursts is unpredictable, accurate parallax measurements will therefore be difficult, except for recurrent systems with a high duty cycle, such as Aql X-1, H1743-322, or GX339-4. Indeed, the feasibility of determining a parallax from the recurrent outbursts of a transient source has recently been demonstrated for the dwarf nova SS Cygni (Miller-Jones et al. Reference Miller-Jones, Sivakoff, Knigge, Körding, Templeton and Waagen2013). The only alternative would be for array sensitivity enhancements to make fainter quiescent systems accessible to VLBI.
Such sensitivity improvements to existing VLBI arrays are either being planned or are already underway, and will both improve the astrometric accuracy of existing measurements and extend the range of possible targets to fainter systems. The recent trend to increasing recording rates (rates up to 2 048 Mbps are now standard) not only improves the sensitivity of a VLBI array, but also permits the use of fainter, closer phase referencing calibrators. This improves the success of the phase referencing process and reduces the astrometric systematics, which scale linearly with calibrator-target separation (Pradel et al. Reference Pradel, Charlot and Lestrade2006). With the option to simultaneously correlate on multiple phase centres at once via uv-shifting, software correlators (e.g. Deller et al. Reference Deller, Tingay, Bailes and West2007, Reference Deller2011) have made it possible to find in-beam calibrators for the majority of low-frequency (≲1.4 GHz) VLBI observations. As well as improving the accuracy of the phase transfer due to the proximity of target and calibrator source, this reduces the slewing and calibration overheads associated with the observation, allowing more time to be spent on the science target.
While these improvements should increase the sensitivity of VLBI arrays by factors of a few, only the large increase in collecting area provided by connecting the SKA to existing VLBI arrays will permit the extension of astrometric studies to a significant number of faint, quiescent systems. Also, by enabling the detection of faint radio emission from extragalactic black holes (either X-ray binaries or ULXs), it could, given a sufficiently long time baseline, allow the measurement of the proper motions of the most luminous black holes in nearby galaxies (or even their ejecta, as tentatively reported for an exceptionally bright transient in M82; Muxlow et al. Reference Muxlow2010). Although the details are still to be determined, a VLBI capability is envisaged in the SKA baseline design, and the main science drivers have been presented by Godfrey et al. (Reference Godfrey2012). The high sensitivity of the SKA could also allow the detection of radio emission from isolated black holes accreting via Bondi-Hoyle accretion from the interstellar medium (Maccarone Reference Maccarone2005), and Fender, Maccarone, & Heywood (Reference Fender, Maccarone and Heywood2013) suggested that astrometric observations could identify such systems via their high proper motions of a few tens to hundreds of milliarcseconds per year, corresponding to velocities of up to several tens of kilometres per second for black holes at a distance of 100 pc.
Increasing the frequency of VLBI observations into the sub-millimetre band improves the resolution and hence the astrometric accuracy of the observations. The advent of the Event Horizon Telescope (Doeleman et al. Reference Doeleman2009), combining existing and planned sub-millimetre telescopes (including the phasing up of the Atacama Large Millimeter Array; see Fish et al. Reference Fish2013) will enable sensitive VLBI observations at millimetre and sub-millimetre wavelengths. With an astrometric precision of a few microarcseconds, this would allow us to resolve the orbits of binary systems, and potentially even detect the thermal emission from the donor star. By tracking the orbits of both components, the system parameters could be constrained with unprecedented accuracy.
Moving from the radio to the optical band, the GAIA astrometric mission (Perryman et al. Reference Perryman2001) promises to revolutionise Galactic astrometry. With the aim of measuring astrometric parameters for 109 stars, complete to V = 20, this mission will measure geometric parallaxes to an accuracy of 11 μas at G = 15, degrading to 160 μas at G = 20. For stars brighter than G = 17–18, its astrometric accuracy will thus rival or exceed that currently achievable with typical VLBI observations. Although most transient X-ray binaries spend the majority of their duty cycles in quiescence with V>20 (e.g. Shahbaz Reference Shahbaz1999), any high-mass or persistent systems (such as Cygnus X-1), as well as the brightest quiescent systems (e.g. V4641 Sgr, 4U1543-47, GRO J1655-40) should be accessible, and we estimate at least ~14 potential targets amongst the known black hole and black hole candidate systems (D. M. Russell, private communication, Reference Russell, Gallo and Fender2013). However, for highly extincted systems in the Galactic plane, VLBI radio observations will remain the astrometric technique of choice.
7 SUMMARY
Astrometric observations are among the most fundamental astronomical measurements. Over the past two decades, high-precision astrometric observations with VLBI arrays have made it possible to determine the proper motions of radio-emitting X-ray binary systems across the Galaxy, and, in a few cases, to determine model-independent distances via geometric parallax. These measurements can probe a range of fundamental physics, from black hole formation mechanisms to the neutron star equation of state, the existence of event horizons, and the spin-powering of black hole jets. Important constraints on system parameters can also be derived from accurate source distances, and even from astrometric residuals, after subtracting the parallax and proper motion signatures from the measured positions.
Such astrometric observations are restricted to systems with detectable, compact radio emission that is causally connected to the central binary system, i.e. X-ray binaries in their hard or quiescent states. For all but the brightest quiescent systems, this requires triggered VLBI observations in the hard X-ray spectral states seen during the rise and decay of X-ray binary outbursts. The proper motion can be measured over the course of a typical few-month transient event, and for recurrent transients, observations over several outbursts can determine the parallax. While high-impact results can be derived from astrometry of individual systems, a large sample of proper motions is required to place useful observational constraints on black hole formation. With only 1–2 black hole X-ray binary outbursts per year (Dunn et al. Reference Dunn, Fender, Körding, Belloni and Cabanac2010), it is therefore important to take advantage of every opportunity to make VLBI observations of X-ray binaries in their hard states, particularly given the sessional nature of both the European VLBI Network and the Australian Long Baseline Array. Although GAIA will significantly extend the sample of X-ray binaries with measured astrometric parameters, the optical faintness of quiescent X-ray binaries and the extinction in the Galactic plane implies that VLBI will continue to play an important role in such astrometric studies.
ACKNOWLEDGEMENTS
JCAMJ acknowledges support from Australian Research Council Discovery Grant DP120102393, and thanks Tom Maccarone, Peter Jonker, and the anonymous referee for insightful comments on the manuscript, and Dave Russell for useful discussions. This work has made use of NASA's Astrophysics Data System.
